Stir together potatoes and boiling water; allow to cool at least 5-10 minutes; meanwhile, sprinkle yeast over warm water; let stand about 5 minutes until foamy.
Stir together potato mixture, yeast mixture, cottage cheese, sugar, oil, egg, and salt; gradually stir in (KitchenAid stand mixer on low speed #1) flour until a soft dough comes together to form a smooth ball.
Knead 9-10 minutes (in KitchenAid stand mixer on medium speed #2) until dough is smooth, elastic, and no longer sticky; add more flour, one tablespoon at a time during the kneading process, if necessary to reduce stickiness.
Place dough into oiled bowl; cover bowl and place in a warm place; allow dough to rise (about 1 1/2 hours) until doubled.
Punch down dough and knead a few strokes to release air; divide dough into 9 (or 12 portions if you want to end up with not-quite-so-large buns) equal portions; with lightly oiled or floured hands, shape each dough portion into a ball, gently pulling and tucking top surface under until smooth.
Place formed buns 2 or more inches apart, tucked sides down, on a greased baking sheet; cover and allow buns to rise (about 30-45 minutes) until puffy.
Bake at 350° for 20-25 minutes, or until buns are light golden-brown; remove from oven and transfer to cooling rack.
NOTE: Makes nine 4 1/2 to 5-inch buns or twelve 3 1/2 to 4-inch buns which, once cooled, may be stored airtight in a plastic bag and refrigerated or frozen if desired.