In a large bowl, combine 2 cups flour, sugar, salt and yeast. In a saucepan, heat milk and butter to 120°-130°. Add to dry ingredients; beat until smooth. Stir in cheese and enough remaining flour to make a soft dough. Do not knead. Cover and let rise in a warm place until doubled, about 45 minutes (dough will be soft).
Punch dough down; divide into 3 portions. Shape each portion into 12 balls. Place in 3 greased 9-in. round baking pans. Cover and let rise in a warm place until nearly doubled, about 30 minutes. Brush with egg white. Bake at 350° for 20-25 minutes or until golden brown.
Test Kitchen Tips
Store these cheddar rolls in an airtight container.