If you want to entertain yourself and you have a car that has a distinct start action (so a real ICE or some hybrids), you can do an easy experiment. Get in the car while it’s off. Press the brake pedal down, fairly firmly, release it, and repeat a few times. If there was residual vacuum while you got in, the first stroke or two will feel normal in the sense that the pedal will move fairly far down without too much effort, but then the pedal will require far more effort to depress farther as you run out of boost.
Now press the brake with moderate force, so it’s as far down as it goes easily in the no-boost state. Maintain that amount of force and start the car. (Do not do this experiment with the gas pedal! You are not trying to move the car!) You should hear the engine start and then, shortly thereafter, feel the pedal move farther down under the same amount of force. That’s the power brakes coming back online.
There’s no electronic magic here — the entire power brake system, on most cars, is entirely mechanical/hydraulic/pneumatic [0] with the possible exception of the antilock brakes and possible traction control. Neither of those will come into play when the car isn’t moving.
[0] Why pneumatic? Energy stored in a compressed or rarified fluid is proportional to the change in volume times the change in pressure. Hydraulics tolerate huge changes in pressure, but hydraulic fluids are highly incompressible, so the volume barely changes and the energy stored is tiny. Hydraulics can transport a lot of energy because then you have the pressure difference times the volume moved, but moving fluid from one cylinder to another does not change total volume. Gasses, on the other hand, allow huge volume changes. This is why it’s very safe to fill all the pipes in your house with water at 80 psi, but pipe manufacturers advise you very strongly not to fill the pipes with compressed air at pressures that high: you will store considerable energy in that air, and, if the pipe fails, it can release that energy rapidly.