Preheat oven to 400 degrees Fahrenheit (205 C). Place dough for pie crust in pie plate or buttered tart pan. Crimp or flute edges if using a pie pan (to crimp: press into pan using table fork or to flute: pinch between thumb and index finger around the edge at the top of the pie plate.) or, if using a tart pan, press dough into fluted edges of tart pan, then trim excess dough by running a rolling pin over the top of the pan. Press a doubled piece of aluminum foil into the dough on the pie plate/tart pan to keep crust in place and bake 10 minutes. (This is called "blind-baking.") Remove from oven and transfer to cooling rack while you prepare filling ingredients. Lower oven temperature to 350 F (177 C). See Notes below to skip the blind baking.
Heat butter and olive oil over medium flame in a 12-inch skillet and sauté onions and broccoli with a pinch each salt and pepper, stirring often until tender – about 10 minutes. Set aside.
Whisk together milk, cream, or half and half, eggs, 1/2 teaspoon salt, 1/4 teaspoon pepper, and Herbes de Provence and set aside. Place ham evenly in bottom of crust and sprinkle with the cooked onions and broccoli. Add grated cheese. Carefully pour the egg mixture on top of the filling.
Bake for 30 -40 minutes on a rimmed baking sheet OR until filling is set, browned, and crust is light gold. A knife inserted at center should come out nearly clean. Do not over bake. Cool at least 10 minutes on rack before cutting and serving hot, warm, or at room temperature. Or cool, wrap loosely, and store for up to 4 days, cut and serve cold.