Heat the oven to 425°F. Toss the broccoli, mushrooms and pepper with 1 tablespoon oil and 1/4 teaspoon salt on a sheet pan; roast for 20 minutes, until browned and no liquid pools underneath.
Reduce the oven to 350°F. Oil a 9-by-13-inch dish with the remaining tablespoon oil and spread the cooked rice evenly across the bottom.
Add the roasted vegetables and feta. Whisk the eggs, milk, remaining salt and pepper and pour over the filling.
Bake for 30 to 35 minutes, until the center reaches 165°F and the egg layer is set.
Rest for 10 minutes before slicing through the rice base. Refrigerate leftovers promptly in shallow containers.