Butter a 12-by-9-inch rimmed baking sheet. Place 1/2 cup warm milk in a small bowl, and sprinkle with yeast; stir to dissolve yeast. Let stand until foamy, about 5 minutes.
In a medium saucepan over medium-high heat, bring remaining 3/4 cup milk just to a simmer. Remove from heat; add 6 tablespoons butter, along with sugar and salt, stirring until butter has completely melted. Set aside.
Place 4 1/2 cups flour in the bowl of an electric mixer. Make a well in the center with your hands, and pour in the yeast mixture, butter mixture, and eggs. Attach bowl to mixer fitted with the dough hook, and beat on low speed until dough just starts to come together, about 2 minutes. Turn out dough onto a lightly floured surface, and knead until smooth and no longer sticky, 5 minutes, adding remaining cup flour as needed. Butter a large bowl; place dough in bowl, turning to coat evenly with butter. Cover with a clean kitchen towel. Let rise in a warm place until doubled in bulk, about 1 1/2 hours. Punch down dough, and let rest 10 minutes.
Melt remaining stick of butter. Divide dough into two equal pieces. On a lightly floured surface, roll out one piece into a 12-by-10-inch rectangle, keeping the second piece covered with the towel. Refrigerate dough until well chilled, about 30 minutes. Repeat with remaining dough. Meanwhile, preheat oven to 375 degrees.
Place dough on a lightly floured work surface with long-side parallel to edge of work surface. Using a sharp paring knife, cut dough vertically into six 2-inch strips. Cut dough horizontally 4 inches from bottom edge of dough. Make another horizontal cut 3 1/2-inches from the first horizontal cut. Make a final horizontal cut 2 1/2 inches from the second cut.
Center a medium rectangle on a large rectangle, and top stack with a small rectangle; repeat with remaining rectangles. Fold each stack in half crosswise and transfer to a buttered muffin tin, folded-side in. Cover with a clean kitchen towel; let rise in a warm place for 30 minutes.
Brush melted butter over top of each roll. Bake until golden brown, about 20 minutes. Brush with more melted butter; serve hot or at room temperature.