Cook potatoes in a large pot of boiling water for about 20 minutes, or until tender. Drain well. Mash with milk and salt. Taste and adjust seasonings if necessary. You should have about 2 cups of mashed potatoes.
Preheat oven to 375 degrees F.
Brush a deep 9-inch pie plate with half the oil. Press in mashed potatoes. Brush with remaining oil.
Bake crust for 25-35 minutes or until potatoes are lightly browned and crusty. Remove from oven, and reduce oven temperature to 350 degrees F.
Arrange broccoli and then cheese over bottom of crust.
Whisk together eggs, milk, salt, pepper and nutmeg. Pour over broccoli and cheese. Sprinkle with green onions.
Bake quiche for 25-35 minutes, or until slightly puffed, browned and just set. Allow to cool for 10 minutes before serving.