In a large bowl, combine 1 cup flour, sugar, salt and yeast.

In a small saucepan, melt the butter over LOW heat. Add the milk and warm until it reaches 120- to 130 degrees F. when tested with an instant read thermometer. Remove from the heat, add to the dry ingredients, and mix with a wooden spoon until smooth.

Stir in the cheese and ½ cup flour. Add just enough of the remaining flour, in small increments, to make a soft, sticky dough (you may not need full amount). Cover the bowl with a clean kitchen towel and let rise in a warm place until nearly doubled, about 45 minutes (dough will be soft).

Meanwhile, grease a 9-inch round baking pan (like a metal cake pan) with softened butter or nonstick cooking spray.

Punch the dough down and with lightly floured hands, shape it into 12 even-sized balls. Tip for even sized rolls - use a bench scraper to cut dough in half and then cut each half in half again for 4 pieces. Then use your hands to shape 3 balls out of each quarter for 12 rolls. Place in a circular pattern in the baking pan, leaving a little space in between the rolls. Cover and let rise in a warm place until nearly doubled, about 30 minutes.

Using a pastry brush, gently brush the rolls to lightly coat with the beaten egg. Bake at 375 degrees F for 12 to 15 minutes or until golden brown. If getting too brown, tent lightly with foil towards the end of the baking time.

Allow to cool in pan for about 5 minutes then serve warm.