Make the dough:
Stir the yeast into the water; let sit for 15 minutes. Combine the flour and salt in the bowl of a stand mixer fitted with the dough hook or in a large bowl. On low speed in the stand mixer or stirring with a wooden spoon, slowly add the yeast mixture and the olive oil alternately to the flour. Knead for 6 minutes on low speed in the stand mixer or on a floured surface by hand until it becomes elastic. The dough should feel soft and just a little sticky. If it feels grainy or dry, add 1 tablespoon warm water at a time (up to 1/4 cup). Knead for another 2 minutes by hand on a floured surface.
Transfer the dough to a lightly oiled bowl that’s at least twice the size of the dough and cover with a damp dishtowel. Let rise at room temperature until almost doubled, about 1 hour. The dough is ready when you poke a finger in it and it holds the impression.
Punch down the dough and divide it into eight 4-ounce balls. Put each ball on a floured surface and, with your hands, flatten and stretch it into a disk that's about 1/2 inch thick. The dough will be fairly elastic and will tend to spring back. Cover each piece with plastic and let rest for 5 minutes. Stretch or roll each disk into an 8- to 10-inch round about 1/8 inch thick (the thinner, the better). If they continue to seem springy and resist rolling, cover and let rest for a few more minutes. Line a rimmed baking sheet with parchment or waxed paper and layer the rounds on it with a sheet of parchment or waxed paper between.
Make the pizza:
Prepare a hot charcoal or gas grill fire. If using charcoal, spread the hot coals across half of the charcoal grate and leave the other half clear. If using gas, turn one of the burners to low to create a cooler zone. Scrub the grill grate with a wire brush and then wipe the grate with a paper towel dipped in oil.
Use both hands to pick up a round of dough. Moving quickly and holding the top edge of the dough, lay the bottom edge (oiled side down) on the hot part of the grill. As soon as the first edge of the dough makes contact with the grill grate, pull gently on the dough as you finish laying it down to stretch it thinly. (If you are able to fit 2 pizzas on the hot part of the grill at the same time, grill 2 at a time.)
Brush the top of the dough with olive oil and sprinkle with salt. Grill without moving until the pizza browns and develops good grill marks on the bottom, 1 to 3 minutes; check frequently to prevent burning.
Using tongs, transfer the dough, browned side up, to a baking sheet and cover the grill. Spread a thin layer of the red pepper purée all over the browned side of the crust. Scatter one-eighth of the roasted garlic and fresh mozzarella over the surface of the pizza. Usig tongs and a wide spatula, slide the pizza, raw side down, back to the hot side of the grill and close the lid. When the bottom has browned and developed strong grill marks, 1 to 3 minutes, move the pizza to the cooler part of the grill.
Close the lid and grill until the pizza toppings are hot to the touch, 3 to 8 minutes more. Check the bottom of the pizza frequently, turning the pizza from back to front and side to side to prevent burning in case your grill has any hot spots. Repeat with remaining dough rounds and toppings. If you want the cooked pizzas to remain hot, put them in a 200°F oven as you finish them.
Scatter the fresh basil evenly over the pizzas and serve.