Pat chicken dry with paper towels. On a plate, mix flour with 1/2 tsp salt and the pepper. Dredge chicken lightly in flour, shaking off excess. This thin coating helps create a golden crust and slightly thickens the sauce later.

Heat olive oil and 1 tbsp butter in a 12-inch skillet over medium-high heat until shimmering. Add chicken in a single layer and cook undisturbed for 5-6 minutes until golden brown. Flip and cook 4-5 minutes more until cooked through (165°F internal temp). Transfer to a plate.

Add remaining 1 tbsp butter to the skillet. Add zucchini and squash, spreading in a single layer. Cook 3-4 minutes without stirring to get some color, then toss and cook 2-3 minutes more until just tender but still holding shape. Season with remaining 1/2 tsp salt. Transfer to the plate with chicken.

Reduce heat to medium. Add garlic and cook 30 seconds until fragrant. Pour in chicken broth and scrape up any browned bits from the pan. Simmer 2 minutes to reduce slightly. Stir in lemon zest, lemon juice, and Parmesan. Cook 1 minute until cheese melts and sauce thickens slightly.

Nestle chicken and vegetables back into the skillet, spooning sauce over top. Simmer 2 minutes to heat through and let flavors meld. Taste and adjust salt or lemon if needed. Sprinkle with parsley and extra Parmesan before serving.