Butter muffin cups with 1 tablespoon melted butter.
Melt 8 tablespoons of butter. Add minced garlic and allow to cool.
Stir together yeast, warm water, and sugar in a large bowl and let stand until foamy, about 5 minutes.
Mix flour, salt, buttermilk, and 6 tablespoons of the garlic butter into yeast mixture with a wooden spoon or rubber spatula until a soft dough forms. Turn out dough onto a well-floured surface and knead, dusting surface and your hands with just enough flour to keep dough from sticking, until dough is elastic and smooth, 6 to 8 minutes. Form dough into a ball.
Put dough in an oiled large bowl and turn to coat. Cover bowl with plastic wrap and a kitchen towel and let dough rise in a draft-free place at warm room temperature until doubled, 1 1/2 to 2 hours.
Punch down dough (do not knead), then halve. Roll out half of dough on a lightly floured surface with a floured rolling pin into a 12-inch square (about 1/8 inch thick; keep remaining half covered with plastic wrap). Brush dough with 1/2 tablespoon garlic butter and cut into 6 equal strips. Stack strips, buttered sides up, and cut crosswise into 6 equal pieces. Turn each piece on a side and put into a muffin cup. Make more rolls with remaining dough in same manner. Separate outer layers of each roll to fan outward. Cover rolls with a kitchen towel (not terry cloth) and let rise in a draft-free place at warm room temperature until doubled and dough fills cups, 1 to 1 1/2 hours.
Preheat oven to 375F with rack in middle. Bake rolls until golden brown, 20 to 25 minutes. Brush tops with remaining 2 tablespoons butter, then transfer rolls to a rack and cool at least 20 minutes.