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.

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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.