In article ,
wrote:

I thought I understood the fuel injection process, but now I'm not so
sure.

When you cold start an IO-360, you are supposed to run the auxiliary
fuel pump with the mixture control knob pushed in until you see a rise
in fuel pressure, then shut off the pump and pull the mixture contol
knob back to shut off, and start the engine, pushing the mixture to
full rich when the engine starts.

Is it a Lycoming? Some have another system.

How does the mixture control shut off the fuel? What does it control?
If the mixture is at idle cutoff, why does the engine start?

Throttle controls air (air box, throttle plate), mixture controls fuel
(needle valve, spider injector controller).
Because excess fuel has been injected into the engine; enough to start
and run fro a few seconds, until the mixture lever is advanced,
allowing fuel to flow, propelled by the engine driven pump.

Another question, why do you start the engine with the mixture at idle
cutoff anyway? Why not start with the mixture at full rich?

Because the engine driven pump would flood the cylinder, creating too
rich a mixture.