Heat grill to medium. Brush a large rimless baking sheet (or an inverted rimmed sheet) with 1 tablespoon oil. On a lightly floured work surface, roll and stretch dough to two 8-inch ovals or rounds. Transfer to prepared sheet. Brush tops with 1 tablespoon oil.
Transfer dough to grill. Cook, without turning over, until undersides are firm and begin to char, 2 to 3 minutes. Using tongs and a large spatula, place crusts on baking sheet, grilled side up.
Scatter crusts with tomatoes, scallion whites, and cheese; season with salt and pepper, and slide back onto grill. Cook, rotating occasionally, until cooked through and cheese begins to melt, 3 to 5 minutes. Transfer to a cutting board.
In a medium bowl, combine spinach, avocado, scallion greens, vinegar, and remaining tablespoon oil; season with salt and pepper, and toss to combine. Pile mixture onto pizzas; halve, and serve.
Flipping a large round of dough on a hot grill can be tricky, so we made two smaller pieces and turned them over with tongs and a spatula -- no sweat.