Cook fettuccine noodles according to package directions, reserve 1/2 cup cooking water, drain and set aside.
Bring large pan to medium high heat and melt 1 tablespoon butter. Season chicken breasts with kosher salt and black pepper and cook for 5-7 minutes on each side, turning once or twice, until browned and juices run clear. Remove from pan, cover with foil and set aside to rest.
Reduce heat to medium and add 1 tablespoon butter to same pan. Add leeks and cook for about 3 minutes or until they begin to soften. Add garlic, diced carrots and diced celery, season with kosher salt and cook 2 minutes, stirring frequently so vegetables soften but don’t brown.
Add chicken stock, bring to boil then reduce to low boil until liquid is reduced by half, leaving about 1 cup.
Add heavy cream and bring to simmer. Reduce heat to medium and cook 5-7 minutes or until sauce is thickened.
Stir in hot sauce and Parmesan and cook 1-2 minutes. Remove from heat and add remaining 2 tablespoons butter. Slice chicken breast into thin slices and add to sauce with fettuccine noodles and toss to coat. Cook another 1-2 minutes. Season with more salt if desired. If sauce is too thick, add reserved pasta water 1 tablespoon at a time until it’s the desired consistency. Garnish with parsley. Serve with additional Parmesan, carrot and celery sticks.