Place 4 leaves of bread on 4 large platters. Pour 1/2 cup hot water over each and allow to soak 3 to 4 minutes, until soft and pliable. Meanwhile, place a nonstick pan over medium heat and add oil. When smoking, crack 4 eggs into pan, keeping separated. Fry eggs sunny side up for 1 to 2 minutes.
Meanwhile, spoon hot tomato sauce over soaked bread and transfer to smaller plates, folding bread to fit on plate. Place 1 egg on each bread, sprinkle with grated cheese and serve.
In a mixing bowl, stir 1 cup flour, 1/2 cup semolina and salt together. In a measuring cup, mix yeast in 1 cup water and stir to dissolve. Mix liquid with flour mixture to form doughy batter. Let sit covered overnight in the warmest place in the kitchen.
Place baking tile in oven and preheat oven to 450 degrees F.
Make a well on a work surface with 2 cups flour and 1/4 cup semolina. Place doughy batter in center and stir in remaining warm water. Mix well and incorporate flour from edges of well until a dough is formed. Bring together and knead 10 minutes, until firm and elastic. Place in a bowl, cover with a towel and let rest 1 hour. Dough should double in size. Cut dough into tennis ball-sized pieces and, using a rolling pin, roll out to 12 to 14-inch circles, about 1/8th to 1/4-inch thin. Place on peel and slide onto oven tiles and bake 2 minutes. Open oven door and turn over. Cook 30 seconds more and remove to board. Bread should be puffed up like a balloon. Immediately cut around edges with a knife to separate top and bottom layers.
Stack the 2 pieces on a kitchen towel and put a box of cereal on top of them to weigh them down. Continue this process until all the dough is cut, baked, separated and stacked. Place each half back into the oven, one at a time, for 15 to 20 seconds, until very crisp. Store wrapped in a brown paper bag. They will last three to four weeks in a dry place.
Saute the onion and garlic in the olive oil over medium heat until translucent, but not brown (about 10 minutes). Add the thyme and carrot and cook 5 minutes more. Add the tomatoes. Bring to a boil, lower the heat to just bubbling, stirring occasionally for 30 minutes. Season with salt to taste.
Serve immediately, or set aside for further use. The sauce may be refrigerated for up to one week or frozen for up to 6 months.
Yield: 6 cups
Recipe courtesy of Mario Batali