I have made two Tesla pancake coils.  The transmitting coil uses a TIP35C transistor with 12 volts power to it.  A base resistor of 470 ohms is required.   I wish someone can show me how and what to use as a MOSFET transistor that would be very efficient and not heat up like the TIP35C does.  A little hand drawn simple circuit would help.
Below is my video showing  my setup and asking for help;

https://www.youtube.com/watch?v=UNa2oHA94iE

Nice build :thumbsup:
As for mosfet's I'd recommend using IRF540 as it's extra low voltage driving the coil and it has a very low RDS(on), which means the powerloss over drain/source will be fairly low as per equation P = I x R^2.
However, this assumes that the transistor works as a switch, with the gate being driven by a PWM like signal for frequency and voltage control.
It looks as though your setup uses the natural self oscillating properties of the coil, in which case the transistor will be working in the active region, which helps explain the seemingly excessive powerloss during operation, which of course heats up the transistor, so there's no going around that in that case I'm afraid, not that I know of anyway.

Just to add a little more info, the transistor you are using is a Bipolar Junction Transistor. MOSFET transistors can be switched more easily but have the same limitation of heating when operating in the "active region". The active region is where the transistor is neither fully on nor fully off - it's in transition. The point of using a pulsed signal is to provide very fast rise and fall times through the active region so it is in that heat producing state for a very short time. There are driver chips that are made for doing that.

So you can use a PWM signal to trigger the mosfet driver chip which will switch a mosfet transistor very quickly so as to have little heat loss from the active region. Most LM555 chip PWM circuits are easy to build on a breadboard and usually have variable resistors to adjust the frequency and duty cycle. So that would allow you to experiment and/or tune the signal as you desire.

Or it could be driven from anything that can output a PWM signal like an Arduino, BASIC Stamp, or any of various micro-controllers.

Tesla patented quite a few circuit controllers to do the same thing but you have to remember he was always working on industrial scale power systems so that would be overkill for what you need.