Preheat oven to 350 degrees. Melt butter in a large saucepan over medium heat. Add the onion and broccoli and saute for a few minutes or until tender. Add the garlic and saute for 30 more seconds. Stir in the ham and pour mixture into frozen pie crust. Top with Cheddar and Feta.
In a medium bowl, whisk together eggs, milk, salt and pepper and pour over fillings in the crust. Sprinkle with smoked paprika. Place pie crust over a cookie sheet and bake for around 45 minutes or until center is set. Cool for 30 minutes before slicing. Slice into 6 wedges.