Preheat the oven to 325°F.
In a large, oven-proof skillet, add the olive oil and butter.
When the butter is melted, add the broccoli, sausage, and onion.
Season with salt and pepper.
Cook for five minutes or until the broccoli is tender.
While the broccoli mixture is cooking, whisk together the eggs and half-and-half.
Sprinkle the cheese over the broccoli mixture, and then pour in the eggs.
Bake for approximately 15 minutes or until the eggs in the center are just set.
Let rest for five minutes before serving.
Garnish with sliced green onion and crushed red pepper, is desired.