Wire the supply through a FET with a low ON resistance and drive the gate from an output of your micro. Fit a double pole single throw push-button, one contact parallel to the FET, the other to an input of your micro.

When the PB is pressed, power is applied toyour circuit and your controller will start. Drive the FET so that it conducts. Now you only have to test the switch input for a new contact closure, then switch off the FET and you're done.

If your want a delayed ON function rather than a delayed OFF, you will need to add some additional circuitry with a capacitance to charge through the "first" contact and a small transistor to drive the FET open once it's above a certain threshhold. Make sure that the FET is driven fast enough!