Dissolve sugar and yeast in warm water in a large bowl; let stand 5 minutes. Lightly spoon flour into dry measuring cups; level with a knife. Add 2 1/2 cups flour, thyme, salt, and egg to yeast mixture; stir until a soft dough forms. Turn dough out onto a floured surface. Knead until smooth and elastic (about 10 minutes); add enough of remaining flour, 1 tablespoon at a time, to prevent dough from sticking to hands (dough will feel slightly sticky). Cover dough; let stand 10 minutes. Knead in Parrano cheese until well combined.
Divide mixture into 12 equal portions, and form each portion into a ball. Place each ball onto a baking sheet sprinkled with cornmeal. Lightly coat tops of rolls with cooking spray; cover with plastic wrap. Let rise in a warm place (85°), free from drafts, 45 minutes or until doubled in size.
Preheat oven to 400°.
Bake rolls at 400° for 18 minutes or until browned on tops and bottoms. Remove rolls from baking sheet; cool completely on a wire rack.