Pour 1 tablespoon olive oil into a large bowl. Set aside. Blend yeast with the warm water and gradually add 1 1/4 cups flour. Knead lightly until dough is smooth and elastic, about 5 minutes. Put in oiled bowl and turn to coat thoroughly. Cover and allow to stand in a warm, draft-free place until doubled in bulk, about 1 hour.
Knead down dough and add 1/4 cup warm water, 3 tablespoons olive oil. Parmesan cheese, remaining flour, basil, and salt. Mix well, then knead dough to a smooth, elastic consistency, adding more flour if necessary.
Preheat oven to 400°F. Spray 1 large or 2 small pizza pans with nonstick cooking spray. Roll dough to fit pan. Brush with olive oil. Bake for 20 to 25 minutes until Focaccia is brown and crisp.
Meanwhile, place tomatoes, remaining olive oil, garlic, and basil in a blender cup. Pulse several times to make a chunky sauce. Salt and pepper to taste. Spoon onto Focaccia crust, top with cheese, and bake until cheese melts, approximately 5 minutes.