Preheat oven to 425 degrees F.
Shape flour into a mound and make an indentation on the top. Pour the oil into the indentation and add the salt. Melt the yeast in a cup of tepid water and pour it into the middle. Start kneading the dough with your hands until all the yeast is absorbed and the dough feels elastic, firm, and looks shiny. See Cook's note*.
Roll it into a ball, sprinkle with flour and place in a bowl. Make a cross on the top. Cover with a cloth and leave it to rise in a warm place for about 1 hour in winter and 30 minutes in summer.
Stretch the dough and then roll it out with a rolling pin on a floury surface, until it is about 1/4-inch thick. Cover the surface of a baking sheet with 1 tablespoon of oil. Sprinkle with sea salt. Place dough on baking sheet and press the edges in firmly with your fingers. Add tomatoes, reserving a little of the juice, and drizzle with olive oil. Cook for 10 minutes. Remove from oven and add chopped mozzarella. Tear the remaining basil leaves and scatter over mozzarella. Drizzle with olive oil. Sprinkle with Parmesan and the rest of the tomato juice. Cook for 5 minutes. Eat immediately.
Cut the tomatoes in half. As you drop them into a medium-sized bowl squeeze them with your hands so that the juice comes out. Add garlic, olive oil, salt, and half the basil leaves. Leave to marinate while the dough rises.
Add more water if dough seems too dry. We find that it's better to use mozzarella which is 2 days old, because it's less watery.