Preheat oven to 425 degrees. Shape the pie crust into a 9-inch springform pan. Line the pan with foil and add pie weights. Place the springform pan on a baking sheet. Bake the crust for 10 minutes. Remove from oven and remove pie weights and foil. Reduce the oven temperature to 325 degrees.
Meanwhile, heat a large saute pan over medium heat. Add a tablespoon of oil and saute the shallots and the broccoli for 5-7 minutes until the broccoli has softened slightly. Season with salt and pepper to taste. Set aside.
In a mixing bowl, whisk together the eggs, heavy cream, salt, and pepper. Set aside.
Layer half of the broccoli and shallots on top of the pie crust. Top with half of the ham and half of the cheese. Repeat layers with the remaining broccoli, shallots, ham, and cheese. Slowly pour the egg mixture over the top.
Cover with foil and bake at 325 degrees for 1 hour and 15 minutes. Remove foil and bake for another 30-45 minutes until the center is mostly set and the edges are golden brown. Let cool for 30-40 minutes before slicing and serving.