Scale out your ingredients for more accuracy into four separate bowls, using small bowls for the salt and yeast, and larger bowls for the water and flour.
Pour the water into a large bowl and gently sprinkle the yeast on top of the water. You’ll start to see it activate as it swells. Let it sit for a few minutes. Pour the flour on top of the yeast & water, and lastly, add the salt. Mix until well incorporated with a wooden spoon. It will be sticky. Let it sit for 20 minutes.
Meanwhile, make your tomato sauce. Pulse 1-pound cherry tomatoes (or 1-14.5-ounce can of no salt added diced or crushed tomatoes) with your olive oil until it’s to your desired consistency. Add ½ teaspoon of sea salt.
With your hands, fold the dough over on itself 8-10 times, turning it around as you fold. It will be sticky and messy, but you want to develop the gluten some. Let that sit, covered with a kitchen towel, for 1½ hours.
If it’s grown considerably and looks bubbly and well risen, you’re ready. If not, give it another 30 minutes. You can refrigerate the dough at this point until you’re about 30 minutes from when you want to eat.
Heat grill to at least 450 degrees or preheat your oven to 450 or hotter (lowest oven rack setting) and place your pizza stone in the grill or oven while it heats. Your stone should be on grill or in the oven at temp for at least 30 minutes.
Working on a generously floured counter, divide dough in half and work with the first half. Shape into a ball with your hands and flatten on the counter, making sure it won’t stick to the counter while it rests. Shape the second half of the dough into a ball, and then gently flatten into a disc shape as well. Place a floured towel over the shaped discs to rest for 30 mins.
Flour a pizza peel or the back of a large cookie sheet. Lift the first disc in your floured hands and spread it gently on the peel, making a bigger circle and forming a crust around the edge by dimpling around the outside with your fingers. Pick up the dough and stretch it over your fists, alternating and letting gravity stretch it, being careful not to let it get too thin in the middle. You want to stretch it to a diameter of about 12 inches before placing the dough onto the pizza peel. Give the dough a little shake on the peel to ensure it will slide off onto the grill easily. If it doesn’t move easily, lift the edges and add more flour underneath.
Spoon the tomato sauce over the dough, followed by the fresh mozzarella, red pepper flakes, fresh basil leaves, and finally, a light drizzle of olive oil and a sprinkle of coarse salt.
Working quickly, slide your pizza onto the edge of the hot pizza stone on the grill or in the oven by giving it a little shake (a spatula can help guide the pizza). Once the edge of the pizza is in line with the edge of the stone, pull the peel away quickly and your pizza should slide fully onto stone.
Watch it closely. Ours took about 8-9 minutes, but it could take up to 16 minutes, depending on the heat intensity.