Preheat the oven to 325 degrees F (165 degrees C). Coat a 13x9-inch baking dish with cooking spray.
Whisk eggs, milk, sour cream, and salt together in a large bowl until well blended. Mix ham, bell peppers, mushrooms, and green onion into egg mixture; pour into the prepared baking dish.
Bake in the preheated oven until a knife inserted into the center comes out clean, about 50 minutes. Cover top of dish with Cheddar cheese; continue baking until cheese is melted, about 3 minutes.
