Cook linguine in water with the tablespoon salt until 1 minute short of the package time. Reserve 1 cup pasta water before draining.
Meanwhile, season cutlets with chicken salt and pepper. Heat oil in a large skillet and cook for 4 to 5 minutes per side until at least 165°F; transfer to a board and rest for 5 minutes.
Lower heat and melt butter in the skillet. Stir garlic and lemon zest for 30 seconds, then add linguine and 1/2 cup reserved pasta water and toss for 1 to 2 minutes until the pasta is tender.
Slice chicken into strips and return with its juices. Remove from heat and toss with lemon juice, Parmesan and parsley.
Add more pasta water a tablespoon at a time if needed to keep the sauce glossy and serve immediately.