Put a rimmed baking sheet on the center rack of the oven and preheat to 375°F (190°C). Brown the sausage in a skillet over medium-high heat for about 6 minutes, breaking it into crumbles. Add the onion and cook 2 to 3 minutes until soft. Drain on paper towels.

Whisk the eggs, heavy cream, salt, and pepper in a bowl until smooth and pale yellow, about 2 minutes.

Place the frozen pie crust on the counter. Scatter the sausage mixture and cheddar in the crust, pour the custard over, and top with Parmesan. Set the quiche on the preheated baking sheet in the oven.

Bake 35 to 40 minutes, until the edges are puffed and golden and the center jiggles slightly. A thermometer inserted in the center should read 170°F (77°C).

Let the quiche cool on a wire rack for at least 10 minutes, then slice and serve warm.