Lightly spoon flour into dry measuring cups; level with a knife. Combine 1/4 cup flour, 1/4 cup water, and yeast in a large bowl, stirring with a whisk; let stand 30 minutes. Stir in 2 cups flour, remaining 1/2 cup water, 1 tablespoon oil, and 3/4 teaspoon salt. Turn dough out onto a lightly floured surface; sprinkle with 1 tablespoon flour (dough will feel slightly tacky). Knead until smooth and elastic (about 8 minutes).
Place dough in a large bowl coated with cooking spray, turning to coat top. Cover and let rise in a warm place (85°), free from drafts, 1 hour or until doubled in size. (Gently press two fingers into dough. If indentation remains, dough has risen enough.)
Prepare the grill to medium-high heat.
Divide dough into 8 equal portions. Working with one portion at a time (cover remaining dough to prevent drying), roll each portion into a 6-inch circle on a lightly floured surface. Place dough circles on a lightly floured baking sheet. Place the dough circles on a grill rack coated with cooking spray, and grill 2 minutes on each side or until golden and crisp.
To prepare topping, combine the vinegar, 1 tablespoon oil, 1/2 teaspoon salt, pepper, and garlic in a medium bowl, stirring with a whisk. Add tomatoes, and toss gently to coat. Spoon 1/2 cup tomato mixture onto each flatbread; sprinkle each flatbread with 2 tablespoons of basil. Serve immediately.