Heat 1 tablespoon of the oil in a medium skillet over medium heat. Add the ham and saute until just starting to brown.
Place the potatoes in a small saucepan and cover with vegetable oil. Cook over low heat until the oil just starts to bubble and the potatoes are tender, about 15 minutes. Remove the potatoes and place on a paper towel-lined plate.
Meanwhile, line a 9-inch pie pan with the prepared pie crust and pierce with a fork. Cover with parchment and fill with the dried beans. Blind bake the crust for 10 minutes. Let cool, then carefully remove and discard the paper and beans.
In a medium bowl, beat together the eggs, half-and-half, salt and pepper. Layer the empty pie shell with the potatoes, ham, and 1/2 cup of the cheese. Pour the egg mixture over the filling. Sprinkle the remaining 1/2 cup cheese and the green onions over the top. Bake on the bottom rack of the oven until the center just sets and the top is golden, 35 to 40 minutes. Let cool 10 minutes before slicing.