Prepare the dough and the sauce per the instructions in Chicago-Style Stuffed Pizza.
Preheat the oven to 450° F/230° C. 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 (7.6 cm) inches larger than the pan and ⅛ inch (3 mm) 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 (2.5 cm). 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 the chanterelle mushrooms. 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 (2.5 cm) 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.