Wood smoke should be going strong at this time (See wood notes that follow).
Bring the assembled pizza on the peel to the hot pizza stone. Maneuver the peel over the pizza stone, and gently shift the peel with a sliding motion to get the leading edge of the pizza crust just off the end of the peel. Touch the crust down to the stone and then quickly and smoothly slide the peel back from under the crust like a magician pulling the tablecloth out from under the place settings.
Close the grill lid and cook the pizza at 500°F for about ten to twelve minutes or until the crust is browned. After the fist five minutes, use a small pizza peel or grilling tongs to rotate the pizza one-half turn (180 degrees) on the stone to help it cook more evenly.
When the pizza is done, remove the pizza from the stone using a pizza peel, or use grilling tongs to lift the edge of the pizza and pull it onto a pizza tray.
Sprinkle with any finishes, like fresh basil, parmesan or finishing salt, then slice and serve.