Mix the flour, yeast, sugar, and salt in the large mixing bowl. Make a well in the flour and add the warm water and olive oil. Mix and knead thoroughly until the dough clings together and cleans the sides of the bowl. Turn the dough out onto a floured work surface and knead for about 5 minutes, until the dough is smooth and soft. Dust with additional flour if the dough is sticking to your hands. Spread some olive oil around the other large mixing bowl and lay the dough in the bottom, turning it over so that the entire surface is coated with a light layer of olive oil. Cover the bowl with plastic wrap and a kitchen towel. Set the bowl in a warm place to rise for about 90 minutes, until doubled.
While the dough is rising, combine all of the sauce ingredients in the 2-quart bowl and stir. Set aside.
Preheat the oven to 450° F. Punch down the dough and turn it out onto a work surface. Knead for about 1 minute. Divide the dough into two pieces, about ⅓ and ⅔ of the dough. Roll out the larger piece in a large circle until it is about 3 inches larger than the pan and ⅛ inch thick. Oil the bottom and sides of the pan. Place the dough into the pan and push it into the bottom and sides. The dough should overlap the pan by about 1 inch. Trim the excess dough with a knife, and set the pan aside.
Roll out the second piece until it is about the same size as the pan. Toss the cheese and spinach together in the medium mixing bowl, and then put it into the pizza pan. Sprinkle with mushrooms and olives. Lay the second piece of dough on top, then crimp the two edges of the dough together with your fingers to form a thick border. Press down on the filling with your hand, and cut a 1-inch slit in the center of the top crust to allow steam to escape. Spread the tomato sauce evenly over the top crust and sprinkle with more grated Parmesan cheese.
Place the pizza pan on the lowest rack for 10 minutes, then move it to the middle rack for 25-30 minutes, until the crust is golden brown. Transfer from the pan to a large wooden cutting board and let it stand for a few minutes before cutting.