Preheat your oven to 375°F, ensuring the rack is positioned in the middle.
Roll out your pie crust on a lightly floured surface using a floured rolling pin.
Fit the rolled dough into the pie plate, allowing any excess crust to hang over the edge.
Prick the bottom of the crust all over with a fork to prevent it from puffing up during baking.
Chill the crust in the refrigerator for about 30 minutes to firm it up.
Line the crust with foil and fill it with pie weights.
Bake until the pastry is set and the edges are pale golden, about 20 to 25 minutes.
Carefully remove the foil and pie weights. Return the crust to the oven and bake until it’s deep golden all over, about 15 to 20 minutes.
As your crust bakes, cook the broccoli in a pot of boiling salted water for about 4 minutes.
Drain and rinse under cold water to stop the cooking process, then pat dry.
Mince and mash the garlic cloves to create a paste with a generous pinch of salt.
In a large bowl, whisk together the garlic paste, eggs, half-and-half, nutmeg, cayenne, and a quarter teaspoon of salt.
Pour the filling into the baked pie shell and gently add the blanched broccoli.
Sprinkle the top with Cheddar and Parmigiano-Reggiano, making sure it's evenly distributed.
Bake the quiche until the custard is just set, about 45 to 50 minutes.
Once baked, cool the quiche for at least 20 minutes before slicing.
Serve your delicious Broccoli Garlic Quiche warm or at room temperature.