In the bowl of a stand mixer, stir yeast into 1 1/2 cups warm water (100° to 110°). Let stand until yeast dissolves, about 5 minutes. Add 1/4 cup oil, the flour, and salt. Mix with dough hook on low speed to blend, then mix on medium speed until dough is very smooth and stretchy, 8 to 10 minutes. Dough will feel tacky.
Cover dough and let rise at room temperature until doubled in bulk, about 1 1/2 hours.
Punch down dough and let rise again until doubled, 30 to 45 minutes. Meanwhile, cut 6 pieces of parchment paper, each about 12 in. long. Prepare a charcoal or gas grill for medium heat (about 350°; you can hold your hand 5 in. above cooking grate only about 7 seconds).
Turn dough out onto a work surface and cut into 6 portions. For each pizza, lay a sheet of parchment on work surface and rub with 1 tsp. oil. Using well-oiled hands, put each dough portion on a parchment sheet. Flatten dough portions, then pat into 9- to 10-in. rounds. If dough starts to shrink, let rest 5 minutes, then pat out again. Let dough stand until puffy, about 15 minutes.
Flip a round of dough onto grill, dough side down. Peel off parchment. Put 1 or 2 more dough rounds on grill. Cook, covered, until dough has puffed and grill marks appear underneath, about 3 minutes. Transfer rounds, grilled side up, to baking sheets. Repeat with remaining dough. (Grilled rounds can stand at room temperature up to 2 hours; reheat grill to continue.)
Arrange your choice of pizza toppings (below) on grilled sides of dough. With a wide spatula, return pizzas, 2 or 3 at a time, to grill and cook, covered, until browned and crisp underneath, rotating pizzas once for even cooking, 4 to 6 minutes.
Make ahead: Complete dough through step 1, then chill, covered, at least 3 hours and up to 2 days (dough will double in size, and flavor will develop as it stands).
Note: Nutritional analysis is for crust per serving.