Preheat the oven to 400 degrees F (200 degrees C).

Bring about 1 inch water to a boil in a pot. Place broccoli in a steamer basket over boiling water and steam until tender but still a bit firm, about 5 minutes.

In a bowl, mix Cheddar cheese, 3/4 cup flour, 1/2 teaspoon salt, and mustard. Mix in 1/4 cup melted butter until evenly moist. Press into the bottom and sides of an 8-inch pie dish.

Melt 1 tablespoon butter in a skillet over medium heat. Add onion and mushrooms and sauté until tender. Stir in 2 tablespoons flour, half-and-half, remaining 1 teaspoon salt, and nutmeg. Bring to a boil and cook for 1 minute. Remove from the heat and mix in steamed broccoli. Gradually stir in eggs. Transfer to the pie crust.

Bake in the preheated oven for 15 minutes. Reduce the heat to 375 degrees F (190 degrees C) and continue baking until a knife inserted in the center comes out clean, about 20 minutes.