Whisk together first four ingredients in a large bowl; slowly whisk in olive oil. Whisk in shallot and 2 teaspoons of the rosemary.
Place tomatoes in a large shallow dish; drizzle with 2/3 cup of the dressing, and marinate until ready to serve.
Place the dough on an unfloured work surface. Knead the remaining 1 tablespoon of the rosemary into the dough. Lightly flour the work surface, and divide dough into 6 equal pieces; shape each one into a 7-inch circle 1/8 inch thick. (If the dough won't stay stretched to 7 inches, let it rest 5 minutes and try again.) Spray both sides of each round of dough with olive oil cooking spray; sprinkle with about 1/8 teaspoon kosher salt. Grill flatbreads over medium-high heat (350° to 400°) 2 to 3 minutes until bubbles appear on top and underside is crisp. Turn, and grill 2 to 3 additional minutes.
Place lettuce in the bowl with the remaining dressing, and toss to coat; arrange on a large platter or six serving plates. Top the greens with the tomatoes and their juices and serve with the grilled flatbreads.