Preheat the griddle to medium heat. Add the thawed tortellini first, cover with a 9×13 foil pan or melting dome and squirt a little water on the griddle under the dome.
Cook 2 to 3 minutes total, flip them around halfway through, add more water, and cover again (if using the foil pan use caution as it will be hot). Set the pan or melting dome aside.
Add some olive oil to the tortellini and to an empty space on the griddle. Put the chicken (if using) and onion over the oil and season with kosher salt, pepper, and Italian seasoning. Cook 2 to 3 minutes, flipping both the tortellini and chicken/ onions a few times with hibachi spatulas.
Add the spinach, tomatoes, and garlic, more kosher salt, pepper, and Italian seasoning. Combine and cook everything together on the griddle for 3 to 4 minutes.
Add the balsamic vinegar, another drizzle of olive oil, mozzarella cheese, and parmesan cheese. Combine with hibachi spatulas until the cheese is melted. Taste for seasoning and add more olive oil, balsamic vinegar, kosher salt, pepper, and/or Italian seasoning.
Serve with more mozzarella cheese and basil sprinkled over top if desired.