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1
Preheat an indoor or outdoor grill to medium heat.
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2
Put the olive oil in a small bowl or measuring cup.
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3
Slice the tomatoes 1/4 inch thick and spread them on a baking sheet or large tray.
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4
Put the scallions on the baking sheet.
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5
Brush everything on both sides with oil and season with salt over the top.
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6
Grill the tomatoes, seasoned-side down, until they develop dark grill marks, 3 to 5 minutes.
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7
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.
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8
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.
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9
Remove the tomatoes and set aside.
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10
Remove the scallions and coarsely chop.
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11
While the tomatoes and scallions are grilling, shape the dough.
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12
Oil two 12- by 15-inch pieces of foil.
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13
Halve the pizza dough.
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14
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.
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15
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.
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16
Be sure to get the edges thin too, or they won't cook through on the grill.
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17
Repeat with the second piece.
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18
Brush the dough with olive oil.
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19
Brush the grill grate with olive oil.
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20
Leaving the dough on the foil, flip them top-side-down onto the grill and peel the foil off.
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21
Cook until the dough just comes up from the grill without sticking, about 3 minutes, and then flip.
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22
Scatter a handful of cheese and then half of the tomatoes and scallions over each pizza.
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23
Top with the rest of the cheese.
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24
Cover the grill and cook, about 6 minutes.
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25
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.
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26
Transfer to a cutting board and serve.
