Prepare your Weber Kettle for pizza cooking – Make sure your bowl and lid vents are open all the way. Fill your charcoal chimney fire starter completely to the top with charcoal briquettes, and when ready empty into the center of the Weber Kettle. With tongs, move the charcoal to the sides of the kettle, creating a ring of charcoal around the kettle, with a circular hole in the middle. Place your grill grate over the coals, and place the pizza stone in the middle of the grill grate, right over this hole. Let your pizza stone heat up for 10 minutes.
Prepare the pizza sauce – Place all of the pizza sauce ingredients into a small mixing bowl and thoroughly mix together.
Assemble the pizza – Drizzle the olive oil over the pizza crust and brush the olive oil all over the pizza. With a tablespoon, apply the pizza sauce to the pizza, completely covering the entire pizza. Add the mozzarella cheese to the pizza in a haphazard yet creative circular pattern. Sprinkle the parmesan-reggiano over the entire pizza. Again, in a haphazard-looking yet creative circular pattern.
Cooking the pizza – Place the pizza onto the pizza stone, put the lid on your Weber Kettle and cook for approximately 10 – 15 minutes. Because the pizza crust is thin, you are only looking for a nice crispy crust, not cooking the dough. The pizza is done when the cheese is completely melted to your satisfaction and the pizza crust is crispy on the sides and bottom.
Remove the pizza from the Weber Kettle – Wearing oven mitts, pick up the pizza stone with the pizza on it and take it to your stovetop to rest for a few minutes. With a metal spatula, move the pizza to a cutting board. Tear fresh basil leaves and scatter them over the pizza, and drizzle olive oil over the top of the entire pizza. Slice the pizza in half, each half in half, and each quarter in half for a total of eight pieces.