Make the dough by adding the flour, yeast, salt, and sugar. Mix. Add the water, and with your wooden spoons, or hands, mix until blended. I use my KitchenAid for this process to keep everything in the bowl during the rising process. Mix for about 30 seconds. The dough should be a bit stiff. Cover the bowl with a towel, and let it sit at room temperature until the dough has doubled in volume, about 2 hours. You can do this ahead of time as well, and place in an oiled container, big enough to house the dough, and place in the refrigerator overnight.
Preheat your oven to 475 degrees, and place your pizza stone, or sheet pan in the oven to preheat. Once preheated, let the stone stay in the oven for about 10 minutes to get nice and hot.
Once risen, divide into two portions. With one of the portions, add some flour to a large pastry board, and take that portion of the dough, and form into a large enough piece to place onto your pizza stone.
Once the pizza stone is hot, carefully place the formed pizza dough onto the stone. Lay on the sauce and evenly spread it onto the pizza dough.
Top with the mozzarella and feta cheese, and evenly distribute that. Top with the chopped gyro meat, and the sliced red onions.
Place in the oven and cook for about 25 minutes or until the crust is nice and golden. Remove and let it sit on an aerated rack, if you have one, for about 5 minutes. Before serving shower the pizza with the fresh, chopped tomatoes and drizzle on the tzatziki sauce. Slice and serve.