Measure the water, yeast, sugar and oil into a food processor and pulse several times to mix. Measure in the flour, cornmeal, and 1 1/2 teaspoons salt. Process until the mixture forms a ball. (If the dough is quite sticky, add 2 to 3 tablespoons more flour and pulse until incorporated. If too dry, add a little dribble of water to the processor.) Transfer to a bowl, cover with plastic wrap and let rise at room temperature while you’re preparing the grill and toppings (at least 20 minutes, or as much as 1 1/2 hours).
Heat a gas grill to medium-high on one section, medium on another, or prepare a charcoal fire distributing the coals so that the fire is hotter (more coals) on one side. Set the grill grates in place. Toss the onion rings with a little oil and put them in a perforated grill pan on the cooler side of the grill. Cover the grill and cook until onions are golden, about 15 minutes. Turn onions occasionally. Remove and break into rings.
Scoop the chicken into a large bowl along with 1 1/2 cups of tomatillo salsa and salt to taste, usually 3/4 teaspoon. Mix to combine.
Punch down the dough and turn out onto a lightly floured work surface. Knead several times to expel as much air as possible. Divide into quarters; flour each piece and your rolling pin. Working with one piece at a time, roll and stretch the dough gently into a 10-inch oval about 1/4-inch thick. Brush or spray the top of the pizza lightly with oil.
Lay an oval of dough, oiled side down onto the grill’s cooking grate on the hotter side. Cover grill, when the bottom has browned, about 4 to 5 minutes, spray the top with oil and flip. Slide onto the cooler side of the grill and spread about 1/4 cup salsa over the top. Evenly scatter on a portion of the chicken, onions, and shredded cheese over the surface of the pizza. Cook until crisp underneath and the cheese has melted, about 4 minutes. Slide the pizza onto a cutting board and cut into serving pieces. Repeat the sequence with the remaining three pizza crusts.