Heat the oven to 350 degrees F. Melt 3 tablespoons of butter in a 10-inch ovenproof omelet pan over medium-low heat. Add the potatoes and fry them until cooked through, turning often, about 10 to 15 minutes. Melt the remaining 5 tablespoons of butter in a small dish in the microwave.
Meanwhile, whisk the eggs, then stir in the ricotta, Gruyere, melted butter, salt, pepper, and basil. Sprinkle on the flour and baking powder and stir into the egg mixture. Pour the egg mixture over the potatoes and place the pan in the center of the oven. Bake the frittata until it is browned and puffed, 50 minutes to 1 hour. It will be rounded and firm in the middle and a knife inserted in the frittata should come out clean. Serve hot.