Place the pie crust in a 9-inch pie plate and crimp the edges. Use a fork to poke holes on the bottom and sides of the pie crust. You can also line the pie crust with parchment paper and add dry beans or pie weights to doubly insure against the crust shrinking.
Bake the crust for 12 minutes. Remove from the oven. If using, use the parchment paper to remove the beans or pie weights. Set the pie plate on a rimmed baking sheet and let the crust cool while you continue the recipe.
Reduce the oven temperature to 375°F.
In a medium nonstick skillet, melt the butter over medium heat. Add the onion and cook, stirring occasionally, until the onion is soft and has reduced in volume by half, 5-6 minutes.
Remove from the heat and spoon the onions into the bottom of the par-baked crust. Top evenly with the diced ham, cheeses, and green onions.
In a bowl or large measuring cup, whisk together the eggs, cream, hot sauce, mustard, salt, and pepper. Slowly pour this mixture evenly over the fillings. Wiggle the pie plate to help the custard settle into the fillings.
Bake for 45-55 minutes or until the center is puffed and set.
Allow the quiche to cool for at least 30 minutes before slicing and serving. Serve warm, at room temperature, or chilled.