Combine the yeast with 1/2 cup warm water in a bowl. Let rest for 10 minutes so the yeast dissolves.
Put the flour into a large bowl with the salt. Make a well in the center of the flour and pour in the dissolved yeast and water mixture. Add 1 cup cool water and the 3 tablespoons of the olive oil. Using a wooden spoon, combine the ingredients until they form one mass of dough. You may need to add some more cool water. Turn the dough onto a lightly floured work surface and knead it for about 10 minutes, until it is silky smooth.
Place the dough into a large, lightly floured bowl and turn it so that the entire surface of the dough is lightly coated with the flour. Cover the bowl with a kitchen cloth and place in a warm area of your kitchen (the top of the refrigerator is ideal), and let rest until it has doubled in size, about 1 1/2 hours.
In the meantime, prepare the topping. In a 12-inch saucepan over low heat, combine the 1/2 cup of the olive oil. Add the scallions, anchovies, sugar and onions and continue carmelizing over low heat until vegetables are tender, about 15 minutes. Add the tomatoes and chili flakes. Stir to combine. Season to taste with salt and pepper, being aware that the anchovies themselves are salty. Continue simmering over low heat, stirring often, for about 10 minutes. Remove from the heat and transfer mixture to a mixing bowl to cool. Allow the mixture to come to room temperature.
When the dough has doubled in size, punch it down and knead for 1 minute in the bowl. On a lightly floured work surface, and using a rolling pin, roll out the dough into a large rectangle, about 1/4 inch thick. Using a sharp knife, cut the rectangle into squares, 3 by 5 inches large.
Fill a deep pot no more than halfway with extra-virgin olive oil. Heat oil over medium-high heat until it reaches a temperature of 350 degrees. The oil should remain at or around this temperature throughout the cooking process. Working in batches of 6, fry the dough until golden brown. Use a slotted spoon or spider to gently drop the dough into the water, being careful not to splatter the hot oil. Using the same spoon or spider, remove the finished bread to a plate lined with paper towels.
As the bread is draining, reheat the tomato mixture, stirring well top recombine the ingredients.
Placed the fried dough pieces on a warm serving platter and spoon about 2 tablespoons tomato mixture over the top of each piece. Sprinkle with caciocavallo cheese, to taste, and serve immediately.