In a mixing bowl, combine 2 cups flour, sugar, yeast and salt. Add water and oil; beat on medium speed for 2 minutes. Add egg and vinegar; beat on high for 2 minutes. Stir in cheese and enough remaining flour to form a stiff dough. Turn onto a floured surface; knead until dough is smooth and no longer sticky, about 2 minutes.
Roll dough to about 1/2 inch thickness. Cut with a 3 inch round cutter. Roll scraps if desired. Coat baking sheets with nonstick cooking spray and sprinkle with 2 Tablespoons cornmeal. Place muffins 2 inches apart on prepared pan. Sprinkle tops with remaining cornmeal. Cover and let rise until doubled, about 1 hour.
Heat an ungreased griddle or electric skillet to 325°F Cook muffins for 20 to 25 minutes or until golden brown, turning every 5 minutes. Remove to wire racks to cool. Split with a fork and toast if desired. Yield: about 16 muffins.