There is an example sketch you can run on an Arduino, and it is (incorrectly and confusingly) named ArduinoISP. It appears in the menu File, Examples, ArduinoISP. When you upload this sketch to an Arduino, the Arduino becomes an ISP programmer. To use the programmer you would go on the Tools, Programmer menu and select "Arduino as ISP"
In the Tools, Programmer menu is another selection called ArduinoISP. This is the programmer you select when you have purchased an ArduinoISP device to use as a programmer. It is a retired or discontinued device but you can still find them from various sellers. https://store.arduino.cc/product/A000092
Thank you very much, dmjlambert, that clears up my confusion on that issue. I'll leave the programmer selection set on "Arduino as ISP" and pretend that the other "ArduinoISP" programmer option doesn't exist. :)
I've already figured out how to burn the bootloader to a new chip and designed a small PCB to do so. (Haven't etched the board yet.)
I'm confused about one other small issue. The instructions I read said to fit a ceramic resonator on the target PCB, connected to pins 9, 10 and ground, yet I just saw another set of instructions on the ADAfruit site that don't mention a resonator, but instead said to connect pin 9 on the target chip to pin 9 on the programming Arduino.
Do you know which is correct, or are both methods acceptable?
(I might need to change my PCB layout.)
Edit: I should add at this point that I haven't actually got my UNO yet - still waiting for it to arrive.