Set the oven to preheat at 375°F. Roll out the pie crust and gently fit it into a 9-inch pie plate, crimping the edges neatly if you prefer. In a spacious bowl, whisk the eggs with milk, salt, and pepper until well combined. Scatter the ham, 1 cup of shredded cheese, and sliced green onions evenly over the prepared crust, then pour the egg mixture over the filling. Top with the remaining ½ cup of cheese. Place in the oven and bake for 35–40 minutes until the center is firm and cooked through. Allow the quiche to rest for 5–10 minutes before cutting and serving.