ultrasonic mist maker
According to the IRFZ44N datasheet  https://www.infineon.com/dgdl/irfz44npbf.pdf?fileId=5546d462533600a40153563b3a9f220d - there is nothing in the circuit that should exceed its maximum ratings.

First thing I'd do is to check the pin orientation of the IRFZ44N  -  https://components101.com/mosfets/irfz44n-datasheet-pinout-features.  If the source and drain are reversed, the internal zener would be forward biased and may be carrying enough current to burn up.  

If D-S are not reversed:- 
Check for wiring errors.
Replace the MOSFET and 555 and remove the 10Ohm resistor.
Check that the MOSFET has stopped burning up. Still burning? - further investigate the MOSFET portion of the circuit especially for shorts to Vcc.
MOSFET not burning? - check the 555 part of the circuit.
When the 555 part OK, replace the 10Ohm resistor. 

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