The circuit can be split in 3 parts.

The first part is the signal generation. Using the two 555 timers a squarewave is generated. The potentiometers allow the dutycycle and frequency to change. I had to use two potentiometers in series on the frequency side since I wanted a larger range but didn't have a higher value potentiometer on hand.

The second part is about driving the current. The generated squarewave from part one is used to switch a transistor. Depending on the current it can get quite hot so I used a big one with a large thermal mass for heat sinking. To limit the current a bit I put a 2.7 Ohm resistor in series with the transformer coil. With another potentiometer the current flowing in the transistors base can be reduced.

The third part is the transformer itself. It is used to step up the voltage on the output and additionally isolates the output from the driving circuit.