Warm 1 tablespoon of olive oil in a 12-inch non-stick skillet over medium-high heat.
Pat the chicken dry with paper towels, then season generously with salt, black pepper, 2 teaspoons Italian seasoning, 1/2 teaspoon garlic powder, onion powder, and lemon zest, tossing to coat evenly.
Add the butter to the skillet along with the olive oil, then place the chicken pieces in the pan and cook for about 3 minutes without moving.
Flip the chicken pieces and continue cooking for another 3 minutes or until they reach an internal temperature of 165 degrees Fahrenheit.
Remove the chicken from the pan and cover loosely with foil to keep warm.
Add the remaining 1 tablespoon of olive oil and 1 tablespoon of butter to the same skillet and heat over medium-high heat.
Toss in the zucchini and yellow squash, season with salt, 1 teaspoon Italian seasoning, and 1/2 teaspoon garlic powder.
Stir occasionally while cooking the veggies until they are just tender, about 4 minutes.
Return the cooked chicken back to the skillet with the vegetables.
Pour the fresh lemon juice over the chicken and veggies and toss everything together gently.
Sprinkle the parmesan cheese and chopped parsley on top, mixing lightly to combine.
Serve immediately, adding extra parmesan and lemon zest on top if you like.