Preheat the oven to 350 degrees F. Butter and flour a 12-by-18-by-1 1/2-inch half sheet pan.
Melt 4 tablespoons of the butter in a 12-inch saute pan over medium-low heat. Add the potatoes and fry them until cooked through, turning often, 10 to 15 minutes. Melt the remaining 8 tablespoons of butter in a small saute pan and set aside.
Meanwhile, whisk the eggs in a large bowl, 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.
Distribute the potatoes over the bottom of the sheet pan. Pour the egg mixture over the potatoes and spread evenly. Bake the frittata until it is browned and puffed, 30 to 35 minutes. Allow the frittata to cool and cut into 20 single-serving squares.
This recipe is adapted from the Potato Basil Frittata recipe published in "Barefoot Contessa Family Style" by Ina Garten, Copyright Clarkson Potter/Publishers 2002.