Dissolve 1 tablespoon sugar in warm water in a small bowl; sprinkle yeast over water. Let stand until yeast softens and begins to form a creamy foam, about 5 minutes.

Combine all-purpose flour, whole wheat flour, and potato flakes in a bowl; set aside. Combine buttermilk, 3 tablespoons softened butter, 2 tablespoons sugar, egg, and salt in a separate large bowl. Add yeast mixture, then add flour mixture, 1/2 cup at a time, stirring until a soft dough forms.

Turn dough out onto a well-floured surface; knead until smooth and elastic, about 3 minutes.

Shape dough into a ball, oil surface with vegetable oil, cover, and let rise in a warm place until doubled in volume, about 1 hour.

Gently punch down dough; divide into 9 equal pieces. Shape pieces into balls. Grease an 8x8-inch baking dish; place dough balls into the prepared dish so they slightly touch each other. Brush tops with about 1 tablespoon melted butter. Cover dish with plastic wrap and allow dough balls to rise in a warm place until doubled in volume, 30 to 45 minutes.

Preheat the oven to 400 degrees F (200 degrees C).

Bake in the preheated oven until golden brown, 25 to 30 minutes.