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Place a pizza stone in the oven. Preheat the oven and stone to 450 degrees for at least 30-45 minutes.
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Divide the dough into two pieces. Flour the counter, and roll one piece of the dough out to about 12 inch diameter.
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Roll two sides up to oppose each other, creating a football shape with a large well in the center. Pinch the edges on either side together. Repeat with the other piece of dough.
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Divide the cheese evenly between the two pieces of dough, filling the centers of each.
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Crack each egg into its own small bowl. Set aside.
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Using a large spatula or cutting board, carefully transfer each piece of dough directly onto the pizza stone.
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Bake for about 12-13 minutes, until the dough browns and the cheese becomes bubbly. Make a small well in the cheese, and carefully pour one egg into the well. Repeat with the other piece of dough.
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Bake for another 2-4 minutes, until the white is almost completely set.
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Remove from the oven. Serve, preferably with a crisp, green salad with a light vinaigrette.