In a small bowl, combine warm water and yeast. Let stand for 5 minutes or until bubbly. Meanwhile, in a large mixing bowl, combine 2-3/4 cups of the flour, the 1/4 cup Parmesan cheese, the basil, sugar, and salt. Add yeast mixture. Stir until combined. Turn dough out onto a lightly floured surface. Knead in enough remaining flour to make a moderately stiff dough that is smooth and elastic (6 to 8 minutes total). Shape into a ball. Place dough in a lightly greased bowl, turning once to grease surface. Cover; let rise in a warm place until double in size (about 1 hour). Punch dough down. Turn dough out onto a lightly floured surface. Divide in half. Cover and let rest for 10 minutes.
Preheat oven to 400 degree F. Lightly grease twenty-four 1-3/4-inch muffin cups. Divide each half of dough into 12 portions. Gently pull each portion of dough into a ball, tucking edges beneath. Arrange balls, smooth sides up, in prepared muffin cups. Cover and let rise in a warm place until nearly double in size (about 30 minutes). Brush tops with milk and sprinkle with the 2 tablespoons Parmesan cheese.
Bake in preheated oven for 12 to 15 minutes or until tops are golden. Remove rolls from pans. Serve warm. Makes 24 rolls.