Bring a large pot of salted water to a rolling boil.
Season the diced chicken with kosher salt, freshly ground black pepper, and the dried oregano–basil mixture.
Heat a large skillet over high heat, spray lightly with cooking spray, then add the chicken and cook 3–4 minutes until no longer pink; remove chicken to a plate.
Cook the spaghetti in the boiling water according to package directions until al dente, reserving about 1/2 cup of the pasta cooking water, then drain.
While the pasta cooks, return the skillet to high heat and add the olive oil. Add the chopped garlic and sauté until golden but not burned, about 1 minute.
Add the halved grape tomatoes to the skillet, season with salt and pepper, reduce heat to medium–low, and sauté about 4–5 minutes until tomatoes soften.
Add the drained pasta to the skillet with the tomatoes and toss to combine; if the pasta seems dry, add some reserved pasta water a few tablespoons at a time until saucy.
Return the cooked chicken to the skillet, add the chopped fresh basil, toss everything together to heat through, and adjust seasoning as needed.
Serve immediately and top with grated cheese if desired.