Warm 1 tablespoon of olive oil in a 12-inch non-stick pan over medium-high heat. Pat the chicken dry with paper towels and season it generously with salt, pepper, 2 teaspoons of Italian seasoning, half of the garlic powder, onion powder, and lemon zest. Toss well to coat the pieces evenly. Add butter to the oil and let it melt, then place the chicken in the pan. Cook for 3 minutes before turning the pieces over, then continue cooking until the chicken is fully cooked through (internal temperature at 165°F), about 3 minutes more. Remove the chicken from the skillet and cover with foil to keep warm. In the same skillet, add the remaining olive oil and butter and heat over medium-high. Toss in the zucchini and yellow squash and season with salt, the remaining Italian seasoning, and the remaining garlic powder. Stir occasionally as the veggies cook until just tender, roughly 4 minutes. Return the chicken to the vegetable mixture and pour in the fresh lemon juice, tossing everything together gently. Finish by sprinkling the parmesan cheese and chopped parsley over the top. Serve right away with extra parmesan and lemon zest if you like.