For the chicken: Pat chicken dry. Season with salt and pepper. In a large resealable bag, marinate chicken in olive oil, garlic and lemon juice for 4 to 18 hours.
Preheat a grill to high. Clean grate.
Make the sauce: Melt the butter in a skillet over medium heat. Add the garlic and lemon zest and cook until the garlic is slightly soft, about 1 minute. Add in the flour and cook, stirring with a wooden spoon, 1 minute. Whisk in the milk and salt and cook, whisking constantly, until just thickened, about 3 minutes. Add the Neufchatel and parmesan cheese; whisk until melted, about 1 minute. Stir in the chopped parsley.
Back at the grill: Turn heat down to medium, and place chicken on the grate. Cook, covered about 7 minutes. Flip and cook an additional 5. Check to make sure your chicken is cooked through
Meanwhile, bring a large pot of salted water to a boil. Add the lemon pasta and cook until al dente, according to manufacturers instructions. Reserve 1 cup pasta cooking water, then drain the pasta and return to the pot.
Add the sauce and 1/2 cup the reserved cooking water to the pasta and gently toss to combine, adding more cooking water as needed to loosen. Season with salt and pepper.
Serve with fresh steamed broccoli and top with additional parmesan cheese and crushed white cheddar Cheez-it’s.