Melt 2 tablespoons butter in skillet over medium-high heat. Add chopped shallots and sauté until tender, scraping up browned bits, about 2 minutes.
Add button mushrooms. Sprinkle with pepper and sauté until liquid evaporates, about 12 minutes.
Add broth, then bourbon. Simmer until liquid thickens and just coats mushrooms, about 14 minutes. While simmering, start the noodles (see below) and combine the sour cream and half-and-half.
Stir in the sour cream mixture and Dijon mustard. Add chicken. Simmer over medium-low heat until meat is heated through, about 2 minutes. Stir in chopped dill. Season to taste with salt and pepper.
Cook noodles in large pot of boiling salted water until tender, about 8 minutes. Drain. Transfer to bowl. Add remaining 2 tablespoons butter and toss to coat. Season with salt and pepper. Divide noodles among plates. Top with chicken and sauce.