Preheat a gas grill or prepare a fire in a charcoal grill.
Divide the dough into 2 pieces. Using a floured rolling pin, roll each piece out to a rough rectangular shape about 12 inches long, 6 to 7 inches wide, and 1/4 inch thick.
Carefully place one rectangle of dough on the hottest part of the grill and cook until the bottom is light golden brown and dry, about 2 minutes. Use tongs or a spatula to carefully flip the dough over and cook on the other side. Transfer to a cutting board and allow to stand for 2 minutes. Repeat with the second piece of dough.
Use a serrated knife to cut each focaccina horizontally in half: Hold the top of the bread with your noncutting hand, maintaining consistent pressure as you carefully cut it into 2 even halves. Smear the garlic paste evenly over the top halves of the bread and divide the provolone evenly between the bottom halves. Sprinkle the rosemary over the garlic halves, and sprinkle the scallions over the cheese halves. Close the focaccina and wrap each one tightly in foil. Grill the sammies, turning once, until they are hot and the cheese is melting, 5 to 6 minutes. Carefully unwrap, cut into 1 1/2-inch-wide strips with the serrated knife, and serve.
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