Heat 1 tablespoon olive oil in a large skillet on medium-high heat until hot. Add sliced zucchini, sprinkle with salt, and cook, on medium-high heat, for about 5 minutes, flipping once, until the zucchini is soft and slightly browned. Remove zucchini from the skillet.
To the same skillet, add one teaspoon of olive oil and sliced chicken breast. Sprinkle the chicken with salt and cook on medium heat for about 5-7 minutes, flipping chicken slices a couple of times, until the chicken is cooked through and no longer pink in the center. Remove from heat, and stir in ¼ cup of basil pesto with the chicken slices.
In a large pot, bring water to boil, add spaghetti, and cook until al dente, according to the package instructions. Drain pasta.
Add cooked and drained pasta to the skillet with the cooked chicken. Add red and yellow cherry tomatoes (each one sliced in half). Add back cooked zucchini. Add another ¼ cup of basil pesto, stir everything together, and reheat gently on low-medium heat. Taste and add extra basil pesto (if desired) and extra salt, if needed. Remove from heat. Sprinkle ⅓ cup of grated Parmesan cheese over the pasta in the skillet.
When serving, sprinkle each individual serving plate with extra Parmesan, if desired.