Heat 1 tablespoon olive oil in a large skillet over medium-high heat until hot. Add the sliced zucchini, sprinkle with salt, and cook for about 5 minutes, flipping once, until the zucchini is soft and slightly browned. Remove the zucchini from the skillet.
In the same skillet, add 1 teaspoon olive oil and the sliced chicken breast. Sprinkle with salt and cook over medium heat for 5-7 minutes, flipping the 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 basil pesto with the chicken.
Bring a large pot of water to a boil. Add the spaghetti and cook until al dente according to the package instructions. Drain the pasta.
Add the cooked and drained spaghetti to the skillet with the chicken. Add the halved red and yellow cherry tomatoes. Return the cooked zucchini to the skillet. Pour in another ¼ cup basil pesto and gently stir everything together. Reheat on low to medium heat. Taste and add extra basil pesto or salt if desired. Remove from heat.
Sprinkle ⅓ cup grated Parmesan cheese over the pasta in the skillet. When serving, add more Parmesan to individual plates if desired.