Preheat the oven to 350 degrees F (175 degrees C).

Place broccoli into a glass bowl. Cover the bowl with plastic wrap. Cook in the microwave in 1-minute intervals until evenly heated through, about 5 minutes; drain and set aside.

Melt butter in a large saucepan over medium heat. Sauté onion and garlic in melted butter until onion is slightly translucent, about 5 minutes. Stir ham and broccoli into onion mixture, then transfer to pie crust.

Beat together cheese, milk, eggs, and black pepper in a medium bowl; pour over ham mixture in crust. Sprinkle with paprika.

Bake in the preheated oven until a knife inserted into the center comes out clean, 30 to 45 minutes.