Preheat oven to 400 degrees F (200 degrees C). Remove pie crust from the refrigerator and allow it to come to room temperature, about 15 minutes.

Unroll the unbaked crust and press it firmly without stretching into the bottom and sides of a glass pie plate. Prick the crust all over with a fork.

Bake in the preheated oven until lightly browned, 11 to 13 minutes. Remove from oven and reduce temperature to 375 degrees F (190 degrees C).

Place broccoli in a microwave-safe bowl and cook in the microwave to desired tenderness, 3 to 5 minutes.

Layer ham, Swiss cheese, and broccoli in the crust-lined plate. Combine eggs, milk, dry mustard, salt, and black pepper in a bowl; pour over broccoli.

Bake in the hot oven until a knife inserted in the center comes out clean, 35 to 45 minutes. Let stand 10 minutes before serving.