Combine ingredients in a stand mixer and mix with dough hook until dough comes together. Mix on medium speed for 5-7 minutes, until dough is smooth and sticky. The dough should clear the sides of the bowl, but should stick to the bottom of the bowl.
Turn dough out onto floured surface. Divide the ball of dough into two equal pieces. Flour your hands and sprinkle each ball of dough with flour, Gently form each piece of dough into a ball, then place on an oiled, parchment-lined baking sheet. Cover with plastic wrap or slip into a large plastic bag. Refrigerate overnight.
Remove dough from refrigerator 2 hours before you plan to bake it. Dust counter with flour, then spray with cooking spray. (I'm not entirely sure what purpose the flour + spray serves, or how it wouldn't get gunky, so I skipped the cooking spray each time.) Place dough balls on the floured surface; coat your hands with flour and sprinkle the dough with flour as well. Press dough into rounds ½-inch thick and 5 inches in diameter. Sprinkle with flour (and spray with oil), and cover loosely with plastic wrap. Let rest for 2 hours.
Preheat your oven to 500F and place your baking rack in the lower third of the oven. Sprinkle your pizza pan lightly with cornmeal. Flour your hands well, and carefully stretch your ball of dough with your hands. [Reinhart says to bounce the dough in a circular motion between your hands, stretching gently with each bounce. Mine stretched tremendously, and eventually tore, as I handled it.] Stretch/toss until dough is 9-12 inches in diameter, then lay on prepared pizza pan. Repeat with other pizza. Top lightly with sauce and toppings.
Bake each pizza for 5-8 minutes on the bottom rack, rotating after 2 minutes if it is cooking unevenly. Wait 3-5 minutes before cutting.