Heat olive oil in a large skillet over high heat until shimmering hot.
Add cherry tomatoes to the skillet in a single layer and let them cook undisturbed for 3-4 minutes until blistered and charred on one side.
Flip tomatoes and cook for another 2-3 minutes until softened and bursting. Season with salt, pepper, and red pepper flakes if using. Remove from heat.
Lay tortillas flat and layer each with mixed greens, shredded chicken, fresh mozzarella, and basil leaves.
Top with blistered tomatoes and drizzle with balsamic glaze.
Fold in the sides of each tortilla and roll tightly from bottom to top.
Cut wraps in half diagonally and serve immediately while tomatoes are still warm.