In a medium bowl, whisk the eggs and heavy cream until light and frothy. Gently stir in the shredded cheddar cheese, salt, and pepper.
Set the mixture aside.
Position a rack in the center of the oven and preheat to 400°F / 200°C.
Heat a 12-inch cast-iron or other oven-safe skillet over medium-high heat.
Add the sausage slices in a single layer and cook, flipping once, until nicely browned on both sides, about 5-6 minutes total.
Add the mushrooms and green onions to the skillet with the sausage, and cook, stirring occasionally, until the mushrooms have released their liquid and are tender, about 5 minutes.
Remove the skillet from the heat.
Pour the prepared egg mixture evenly over the sausage and vegetables. Carefully transfer the skillet to the preheated oven.
Bake for 20-25 minutes, or until the center is just set and the edges are slightly puffed and golden.
Let the skillet cool for a few minutes before serving. Run a spatula or knife around the edge to loosen.
Garnish generously with extra green onions, if desired, and serve warm directly from the skillet. Leftovers can be stored in an airtight container in the refrigerator for up to 4 days.
