Prepared pizza dough is suppler at room temperature than when chilled, but also a bit stickier. Working on an oiled piece of foil solves the problem, and allows you to easily flip the dough onto the grill. Make sure you grill your tomatoes until tender and cooked through: Super-green ones take a bit of time to soften up while those turning toward yellow or just beginning to blush will get soft and juicy pretty quickly. If you have some bacon on hand, grill it up and crumble it over the tomatoes before the last layer of cheese goes on.
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.
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