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