1/3 cup olive oil
6 medium to large green tomatoes (about 3 pounds)
1 pound store-bought pizza dough, at cool room temperature
8 ounces sharp white Cheddar, grated (about 2 /12 cups)
Preheat an indoor or outdoor grill to medium heat. Put the olive oil in a small bowl or measuring cup. Slice the tomatoes 1/4 inch thick and spread them on a baking sheet or large tray. Put the scallions on the baking sheet. Brush everything on both sides with oil and season with salt over the top.
Grill the tomatoes, seasoned-side down, until they develop dark grill marks, 3 to 5 minutes. Flip, brush with more oil, season again and cook until charred and tender when pierced with the tip of a paring knife, another 3 to 5 minutes. Put the scallions on the grill and grill, flipping once, until charred in spots (don't overdo it) and tender when pierced in the white part, about 3 minutes per side. Remove the tomatoes and set aside. Remove the scallions and coarsely chop.
While the tomatoes and scallions are grilling, shape the dough. Oil two 12- by 15-inch pieces of foil. Halve the pizza dough. Drape one half over your fist and gently tug and turn it, letting the weight of the edges help stretch it thinner, so that it makes a rough oblong. Then put it on the oiled foil and press, pull and push gently until it is very thin, and measures about 9 by 12 inches. Be sure to get the edges thin too, or they won't cook through on the grill. Repeat with the second piece.
Brush the dough with olive oil. Brush the grill grate with olive oil. Leaving the dough on the foil, flip them top-side-down onto the grill and peel the foil off. Cook until the dough just comes up from the grill without sticking, about 3 minutes, and then flip. Scatter a handful of cheese and then half of the tomatoes and scallions over each pizza. Top with the rest of the cheese. Cover the grill and cook, about 6 minutes. Give the pizzas a 45-degree turn, cover and cook until the bottoms are deep golden brown, the cheese is bubbling and the dough is cooked through, another 6 minutes. Transfer to a cutting board and serve.