Preheat oven to 425°.
Cut onion into 8 wedges, leaving root intact. Place onion wedges, shallots, and garlic in a 13 x 9-inch baking dish. Combine balsamic vinegar, butter, honey, and thyme in a small bowl, and drizzle over onion mixture, tossing to coat. Bake at 425° for 30 minutes or until tender, stirring occasionally.
Line a baking sheet with parchment paper, and secure with masking tape. Sprinkle paper with cornmeal. Unroll dough onto paper; let stand 5 minutes. Pat dough into a 14 x 12-inch rectangle. Arrange onion mixture over dough, leaving a 1-inch border. Sprinkle evenly with salt and pepper. Fold edges of the dough over onion mixture (dough will not completely cover onion mixture). Sprinkle with cheese.
Bake at 425° for 15 minutes or until lightly browned. Cool on baking sheet 10 minutes on a wire rack. Sprinkle with parsley.