Good news first: bootloader burned! 
I was strongly suspecting that the TFT was sucking current in the exact moment when the pro mini is reset. In fact, the default behaviour of the TFT is to show a blank white screen when it is turned on. White color drains a lot of current!
So I completely detached the pro-mini from the breadboard. Then I connected all the wires from the USBasp to the arduino pins.
BOOM!
The "Burn bootloader" command (with Bootloader=No option) has worked like a charm:
/home/giulio/.arduino15/packages/arduino/tools/avrdude/6.3.0-arduino9/bin/avrdude -C/home/giulio/.arduino15/packages/MiniCore/hardware/avr/2.0.0/avrdude.conf -v -patmega328p -cusbasp -Pusb -Uflash:w:/home/giulio/.arduino15/packages/MiniCore/hardware/avr/2.0.0/bootloaders/empty/empty.hex:i -Ulock:w:0x0f:m
avrdude: Version 6.3, compiled on Jan 17 2017 at 11:00:16
Copyright (c) 2000-2005 Brian Dean, http://www.bdmicro.com/
Copyright (c) 2007-2014 Joerg Wunsch
System wide configuration file is "/home/giulio/.arduino15/packages/MiniCore/hardware/avr/2.0.0/avrdude.conf"
User configuration file is "/home/giulio/.avrduderc"
User configuration file does not exist or is not a regular file, skipping
Using Port : usb
Using Programmer : usbasp
AVR Part : ATmega328P
Chip Erase delay : 9000 us
PAGEL : PD7
BS2 : PC2
RESET disposition : dedicated
RETRY pulse : SCK
serial program mode : yes
parallel program mode : yes
Timeout : 200
StabDelay : 100
CmdexeDelay : 25
SyncLoops : 32
ByteDelay : 0
PollIndex : 3
PollValue : 0x53
Memory Detail :
Block Poll Page Polled
Memory Type Mode Delay Size Indx Paged Size Size #Pages MinW MaxW ReadBack
----------- ---- ----- ----- ---- ------ ------ ---- ------ ----- ----- ---------
eeprom 65 20 4 0 no 1024 4 0 3600 3600 0xff 0xff
flash 65 6 128 0 yes 32768 128 256 4500 4500 0xff 0xff
lfuse 0 0 0 0 no 1 0 0 4500 4500 0x00 0x00
hfuse 0 0 0 0 no 1 0 0 4500 4500 0x00 0x00
efuse 0 0 0 0 no 1 0 0 4500 4500 0x00 0x00
lock 0 0 0 0 no 1 0 0 4500 4500 0x00 0x00
calibration 0 0 0 0 no 1 0 0 0 0 0x00 0x00
signature 0 0 0 0 no 3 0 0 0 0 0x00 0x00
Programmer Type : usbasp
Description : USBasp, http://www.fischl.de/usbasp/
avrdude: auto set sck period (because given equals null)
avrdude: warning: cannot set sck period. please check for usbasp firmware update.
Writing | ################################################## | 100% 0.00s
avrdude: 1 bytes of lock written
avrdude: verifying lock memory against 0x3f:
avrdude: load data lock data from input file 0x3f:
avrdude: input file 0x3f contains 1 bytes
avrdude: reading on-chip lock data:
Reading | ################################################## | 100% 0.00s
avrdude: verifying ...
avrdude: 1 bytes of lock verified
avrdude: reading input file "0xfd"
avrdude: writing efuse (1 bytes):
Writing | ################################################## | 100% 0.01s
avrdude: 1 bytes of efuse written
avrdude: verifying efuse memory against 0xfd:
avrdude: load data efuse data from input file 0xfd:
avrdude: input file 0xfd contains 1 bytes
avrdude: reading on-chip efuse data:
Reading | ################################################## | 100% 0.00s
avrdude: verifying ...
avrdude: 1 bytes of efuse verified
avrdude: reading input file "0xd7"
avrdude: writing hfuse (1 bytes):
Writing | ################################################## | 100% 0.01s
avrdude: 1 bytes of hfuse written
avrdude: verifying hfuse memory against 0xd7:
avrdude: load data hfuse data from input file 0xd7:
avrdude: input file 0xd7 contains 1 bytes
avrdude: reading on-chip hfuse data:
Reading | ################################################## | 100% 0.00s
avrdude: verifying ...
avrdude: 1 bytes of hfuse verified
avrdude: reading input file "0b11110111"
avrdude: writing lfuse (1 bytes):
Writing | ################################################## | 100% 0.01s
avrdude: 1 bytes of lfuse written
avrdude: verifying lfuse memory against 0b11110111:
avrdude: load data lfuse data from input file 0b11110111:
avrdude: input file 0b11110111 contains 1 bytes
avrdude: reading on-chip lfuse data:
Reading | ################################################## | 100% 0.00s
avrdude: verifying ...
avrdude: 1 bytes of lfuse verified
avrdude done. Thank you.
avrdude: AVR device initialized and ready to accept instructions
Reading | ################################################## | 100% 0.01s
avrdude: Device signature = 0x1e950f (probably m328p)
avrdude: NOTE: "flash" memory has been specified, an erase cycle will be performed
To disable this feature, specify the -D option.
avrdude: erasing chip
avrdude: auto set sck period (because given equals null)
avrdude: warning: cannot set sck period. please check for usbasp firmware update.
avrdude: reading input file "/home/giulio/.arduino15/packages/MiniCore/hardware/avr/2.0.0/bootloaders/empty/empty.hex"
avrdude: writing flash (2 bytes):
Writing | ################################################## | 100% 0.13s
avrdude: 2 bytes of flash written
avrdude: verifying flash memory against /home/giulio/.arduino15/packages/MiniCore/hardware/avr/2.0.0/bootloaders/empty/empty.hex:
avrdude: load data flash data from input file /home/giulio/.arduino15/packages/MiniCore/hardware/avr/2.0.0/bootloaders/empty/empty.hex:
avrdude: input file /home/giulio/.arduino15/packages/MiniCore/hardware/avr/2.0.0/bootloaders/empty/empty.hex contains 2 bytes
avrdude: reading on-chip flash data:
Reading | ################################################## | 100% 0.07s
avrdude: verifying ...
avrdude: 2 bytes of flash verified
avrdude: reading input file "0x0f"
avrdude: writing lock (1 bytes):
Writing | ################################################## | 100% 0.01s
avrdude: 1 bytes of lock written
avrdude: verifying lock memory against 0x0f:
avrdude: load data lock data from input file 0x0f:
avrdude: input file 0x0f contains 1 bytes
avrdude: reading on-chip lock data:
Reading | ################################################## | 100% 0.00s
avrdude: verifying ...
avrdude: 1 bytes of lock verified
avrdude done. Thank you.
After that, I uploaded my sketch (32726 bytes) with success by using the command "Upload Using Programmer"
Now the bed news.... 
After placing the pro-mini back to the breadboard and powering it......... nothing. It doesn't start. The TFT screen stay blank and even pressing the reset button can resuscitate the arduino.
I'm going crazy....