Preheat the oven to 350 degrees. Heat the butter and olive oil in a skillet over medium high heat. Add the mushrooms and stir to coat. Let them cook without stirring for a few minutes until they start to release moisture and brown. Stir and brown on the other side. Move the mushrooms to one side of the skillet, add another splash of oil and the onions, zucchini, and squash. Season with salt and pepper and cook, stirring occasionally, until browned and softened just slightly.
Meanwhile, whisk eggs, cream or milk, 1 C of cheese, and spices in a large bowl. Spread the veggies into an even layer in the skillet, pour the egg mixture evenly over top, then scatter the remaining cup of cheese over everything.
Let cook on the stove just until the edges begin to set, then transfer to the oven.
Bake until the center is mostly set, about 18 minutes. Enjoy hot or room temp!
If you’re using a 10″ skillet, use 10 eggs. If you’re using a 12″ skillet, use 12 eggs.