Preheat oven to 425°F. Coat a 9-inch glass pie pan with cooking spray.
Toss 3 cups broccoli with 1 tablespoon oil on a large rimmed baking sheet until coated; arrange in an even layer. Roast until tender and lightly charred, about 15 minutes. Reduce oven temperature to 375°F.
Meanwhile, whisk 6 eggs, ½ cup fontina, ⅔ cup Parmesan, ½ cup half-and-half and 1 teaspoon rosemary in a medium bowl until combined. Fold in rinsed beans and the roasted broccoli.
Transfer the egg mixture to the prepared pie pan. Sprinkle with the remaining ½ cup fontina.
Bake until the quiche is set and the edges are golden brown, 30 to 34 minutes. Let stand for 10 minutes before slicing.
