Preheat oven to 425 degrees F
Sprinkle Swiss cheese, finely chopped broccoli, green onions, and ham over bottom of the pie crust. Place pie crust on a baking sheet to protect against any spills.
In a separate bowl, whisk together eggs, milk, salt, and pepper until it resembles a light butter color.
Pour into the pie crust. Place baking sheet with the quiche on top into the oven uncovered and bake for 15 minutes.
Reduce heat to 325 and bake another 30-35 minutes, until the eggs are just firm.
Remove from oven and let rest for 10 minutes. Slice and serve!