NE555 Controlled Transformer Voltage Spikes
You should try driving your transformer with a half-bridge followed by a capacitor.




The left graph is the node between the two transistors.
The middle graph is left "out".
The right graph is the right "out". 
The CLK source is the output of your NE555 (the node of M1's gate in your schematic).


This way the drains of the half-bridge transistors will stay within sane voltages while the node between the capacitor and inductor are a little bit higher. And finally you get a 1 kV across your 4 MΩ resistor.

Here is the link to the simulation in case you want to play around. 



In order to drive a LC tank, you have to match the resonating frequency of the LC tank. 

This formula can be used to calculate the oscillating frequency in Hz if capacitance and inductance is known:

$$f=\frac{1}{2\pi \sqrt{LC}}$$

This formula can be used to calculate the capacitance required if the frequency (in Hz) and inductance is known:

$$C=\frac{1}{L(2\pi f)^2}$$

Because you will be oscillating at the resonance frequency, magical things will happen. In an ideal world with ideal components, the voltage across \$R_5\$ would go into the 200 kV range. In the simulation that does not happen because the transistors can't source/sink enough current to make that happen. So in your case you might get lower/higher voltage than 1 kV, it really depends on the \$r_{d(on)}\$ of your transistors and the ESR of the capacitor and transformer. 

If you are getting too high voltages across \$R_5\$, then you can decrease it by increasing the size of the capacitor, and recalculate the resonance frequency. And vice versa if you are getting too low voltage, just reduce the capacitance value and recalculate the resonance frequency. You can also reduce the voltage across \$R_5\$ by adding a small series resistance with the capacitor so less energy will be transferred through the transformer.



Feedback regarding your schematic:

I am not sure why you are using a 100 Ω in series with your NE555. I'd throw that away. I'd also reduce R4 to something like 20 Ω.