It's either running out of fuel (likely) or spark.
Loosen the gas cap for a moment and see if you are good for another 10 minutes. This is typical of a venting issue.
Change the fuel filter. It may be partially restricted enough, that the carb bowl slowly runs out of gas and the engine dies. Mower sits a few minutes and carb bowl SLOWLY refills. Repeat....
When you engage the blades, the engine requires more fuel. (governor opens the throttle plate on the carb & more air/fuel is drawn in)
You are probably restarting before the carb has refilled "enough".