Generally, to prevent over-discharge of the battery, you use a battery protector module (which uses vow-voltage FETs).
Since most of the logic is CMOS, each of the attached devices really should use no power when not active.
A 3.3 V 8 MHz Pro Mini - with regulator and pilot LED removed - will run just fine on 4.2 V.
Boost converters are interesting. They have a "disable" input but it merely shuts off the boost function so that the output voltage becomes the same as the input.
I think I know what kind of battery protector module you're talking about, and I did consider it, but turns out, it takes up way too much space in my robot (it is very small, 12x10 cm). So I thought I could integrate that functionality myself, since I still have one I/O pin for switching the MOSFET, and one analog I/O pin for measuring voltage (before the boost converter, of course).
Maybe I'll try the Pro Mini next time, but I have a bunch of Nanos lying around, so I designed my circuit around it specifically. That mini USB port is very convenient for uploading new code to the robot once it's complete.
One of the problems of asking questions on here is deciding whether to post your entire project, which is helpful in that the rest of us can see what you are doing and might be able to offer useful advice beyond what your immediate problem, but then makes for an awful lot to read through, or post the minimum that exhibits the unwanted behaviour as you did, in which case it's difficult to offer more general advice.
I have a UPS system in my house to keep the lights on when there's a power cut. It has a 50V power supply, a 50V 12Ah lead acid battery and an inverter. Originally it had no control, so if the mains when off it would discharge the battery completely if left alone. Lead acid batteries can withstand some abuse so I wasn't too concerned. LiPos are a lot more fussy. I did however build a control system that disconnects the load if the batteries become too much discharged and I then had your problem. After some experimenting I realised that once the load was disconnected there was enough charge in the batteries to keep the controller running for ages before that would need to be disconnected too. As a result I have a phased disconnection, the last thing to be disconnected is the controller. As has been commented by others, this cannot be done with 1 transistor of any kind, my control circuit uses 2 MOSFETs.
I hope that helps a bit.
I know!! Still, I thought I'd take the challenges on at the time, which is why I described just a specific part of my project that I can't get to work. Besides, in my experience, telling too much about the project often results in derailed topic, as people start pointing out problems/suggestions about other parts of that project, then some people get into arguments, and in the end, I still don't have the answer to the original question So I'm trying to keep it simple.
I will try this setup with 2 MOSFETs, or a MOSFET and a BJT. Not sure which is better, some examples feature 2 MOSFETs, while others have one MOSFET and one BJT. I guess both would work.