Bring milk to a boil in a large sauce-pan over medium heat; stir in grits, and cook, stirring often, 5 minutes (mixture will be very thick). Remove from heat; add salt and cheese, stirring until cheese is melted. Let stand 25 minutes, stirring occasionally.
Combine 1 cup warm water, sugar, and yeast in the mixing bowl of a heavy-duty stand mixer; let stand 5 minutes. Add grits mixture, beating at medium-low speed with the dough hook attachment until well blended.
Add 4 cups flour, 1 cup at a time, beating until blended after each addition and stopping to scrape down sides as necessary. Gradually add enough flour to make a stiff but slightly sticky dough. Dough will form a ball around mixer attachment.
Shape dough into a ball with well-floured hands, and place in a well- greased bowl, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in bulk.
Punch dough down, and divide into thirds; roll each third into a 14- x 9-inch rectangle on a lightly floured surface. Spread basil pesto evenly over each dough rectangle; sprinkle each with mozzarella or crumbled goat cheese. Roll up, jelly-roll fashion, starting with each short side and ending at middle of dough; form 2 rolls per loaf. Place into lightly greased 9- x 5-inch loafpans; cover and let rise in a warm place (85°), free from drafts, 45 minutes or until doubled in bulk.
Bake at 350° for 35 to 40 minutes or until golden. Let bread cool in pans on wire racks 10 minutes. Remove from pans, and cool completely on wire racks.
Note: For testing purposes only, we used Cracker Barrel Vermont Sharp White Cheddar.