Cook fettuccine in water with the tablespoon salt until 1 minute short of the package time. Reserve 1 cup pasta water before draining.
Meanwhile, season chicken with its salt and pepper. Heat oil in a large skillet and cook cutlets for 4 to 5 minutes per side until at least 165°F; transfer to a board to rest.
Reduce heat to low and melt butter in the skillet. Add cream and milk and bring to a gentle simmer for 2 minutes, scraping up the lightly browned bits.
Add fettuccine and 1/4 cup reserved pasta water and toss for 1 to 2 minutes until the pasta is tender. Remove from heat and stir Parmesan in gradually, loosening with more water if needed.
Slice the rested chicken and serve over the sauced fettuccine while the coating is smooth and loose.