Preheat an outdoor grill for medium heat and lightly oil the grate.
Brush the surface of each flatbread round with olive oil. Flip over and spread 1 tablespoon spinach pesto and 1/4 cup mozzarella cheese over each.
Place flatbreads oiled-side down on the grill. Cook until cheese melts, 2 to 3 minutes. Place flatbreads on a heat-proof plate. Top each with 1 cup romaine lettuce, 2 1/2 ounces grilled chicken, 1/4 of the tomatoes, and 1 teaspoon Parmesan cheese.
