Grease and lightly flour a 9 x 13 baking pan. In a food processor, combine 2 cups flour, 1/2 cup powdered sugar, and the butter. Pat evenly into pan. Bake in a preheated 350 degree oven for 15 minutes. Reduce heat to 325.
Beat eggs in a medium bowl. Add 2 cups sugar, 4 tablespoons flour, baking powder, Key lime juice, coconut and lime zest. Stir until well-mixed. Pour mixture over baked crust.
Bake in preheated 325 degree oven until top is firm, about 35 minutes. Let cool slightly. Sprinkle with powdered sugar and toasted coconut.