In a bowl, mix all-purpose flour, whole-wheat flour, baking powder, baking soda, sugar, and salt.
In another bowl, whisk to blend eggs, buttermilk, and 3 tablespoons butter. Add to dry ingredients and stir just until batter is evenly moistened.
Ladle batter, 1/4 cup at a time, onto a lightly buttered medium-hot (350°) griddle or lightly buttered 10- to 12-inch nonstick frying pan (or use 2 pans at the same time) over medium heat. Scatter 3 to 5 banana slices onto each pancake and cook until edges of the cake look dry, 3 to 4 minutes. Turn with a wide spatula and cook until browned on the bottom, 3 or 4 minutes longer.
Serve pancakes immediately, or place in a single layer on baking sheets and keep warm in a 200° regular or convection oven until all are cooked.