Preheat oven to 375 degrees F (190 degrees C). Lightly grease a 9 inch pie pan.
Press potatoes into greased pie plate. Brush with melted butter. Bake in preheated oven until lightly browned, about 10 to 15 minutes.
In a large bowl, stir together beaten eggs, milk, cheese, ham, salt and pepper. Pour egg mixture into baked crust.
Bake in preheated oven until center is set, about 20 minutes. The quiche will be browned on top and a knife inserted into the center will come out clean.
