Bring about 1 inch water to a boil in a pot. Place the chopped broccoli in a steamer basket over boiling water, and steam 5 minutes, or until tender but firm.
In a bowl, mix the Cheddar cheese, 3/4 cup flour, 1/2 teaspoon salt, and mustard. Mix in 1/4 cup butter until evenly moist. Press into the bottom and sides of an 8 inch pie dish.
Melt the 1 tablespoon butter in a skillet over medium heat, and saute the onion and mushrooms until tender. Stir in 2 tablespoons flour, half and half, 1 teaspoon salt, and nutmeg. Bring to a boil, and cook 1 minute. Remove from heat. Mix in steamed broccoli. Gradually stir in eggs. Transfer to the pie crust.
Bake 15 minutes in the preheated oven. Reduce heat to 375 degrees F (190 degrees C), and continue baking 20 minutes, until a knife inserted in the center comes out clean.
