Step 12: Oscillators!
In order to obtain 225W at 12kV we need 54mA, and assuming a 100% efficiency we’d likewise need to draw 6.25A from a 36V source. Of course 100% efficiency is unobtainable, but 10A at 36V is still not unreasonable for a hobbyist to supply.
A choice oscillator for this task is the current-fed ZVS Oscillator; an LC resonant zero-voltage switching circuit. Although a Hartley or a Colpitts oscillator could be used, both are not switched under no-load conditions, and would thus burn significant amount of power when the MOSFET travels through its linear region.