The inverter is doing all the work - the circuit is essentially a strange shaped buck controller. What it's doing is building up energy in the inductor and dumping it all at once into the LED.

The microcontroller is there as a programmable pulse-width generator. Yes, you can do this with a NE555, at the cost of much more power and more components, and they don't like low duty cycle and are less temperature-stable.

Having put the microcontroller on there, you must also put on its required decoupling capacitors. You might be able to remove them in a situation this simple with a bit more testing.

MOSFET Q1 is required because the peak current is quite high in the inverter, more than 20ma. It also helps to steer the inductor current into the LED rather than the protection diodes of the PIC. Note the LED is "upside down".

MOSFET Q2 and the resistor are a convenience to prevent the circuit being destroyed by inserting the battery the wrong way up.

More at https://hackaday.io/project/11864-tritiled/log/62875-v22-rel...

Check out the hackaday project page for more detail https://hackaday.io/project/11864-tritiled I should make clear this is not my project.