Heat the butter and oil over medium heat in a 10 inch cast iron skillet. Saute shallots until soft, about 6 minutes. Add the artichoke hearts and tarragon. Season to taste with salt and pepper.
Meanwhile, beat eggs and milk together in a mixing bowl. Pour over shallots and artichokes. Let cook for about 5 minutes, or until edges are just set. sprinkle the top with cheese and bake in oven for 8 minutes; it should just be set throughout.