Check if your pilot flame sufficiently heats thermopile during operation and make sure also that it is free of soot or any form of dirt, thermopile generates small amount of electricity to energize pilot and main valve of the gas valve for pilot and main burner operation. This electricity passes through hi limit mechanical contacts. Hi limit on fryers act as safety device which will shut down pilot and burner flame when oil temperature reaches 450F.
If your fryer is set at 350F and over shoots up to 450F (which will trip hi limit and shut down pilot and burner flame), the thermostat is not good, but, if hi limit trips on temperature much lower than 450F, hi limit is bad.