Pat the chicken dry with paper towels and season it generously with salt and pepper.
Melt the butter over medium heat in a large deep pan or Dutch oven. Add the chicken pieces skin-side down and cook for 10-12 minutes, occasionally turning, until all sides have browned. Transfer chicken to a plate and set aside. NOTE: The chicken will not be cooked through.
Add the shallot and garlic to the same pan and cook for 3-4 minutes until the shallots begin to soften. Add the sausage and continue cooking until it begins to brown.
Sprinkle flour into the pan. Add wine and cook for 3-4 minutes, scraping up any browned bits from the bottom of the pan. When the wine mixture has reduced slightly, stir in the chicken broth. Return the chicken to the pan, nestling it down into the broth.
Add the heavy cream to the pan and season with salt and pepper to taste. Simmer at medium heat for 45 minutes or until the chicken is cooked through.
While the chicken is cooking, trim Brussels sprouts and cut them in half. Heat a dry pan or skillet over medium heat and cook the Brussels sprouts until they get nicely browned but not charred.
Ten minutes before the chicken is finished cooking, add the Brussels sprouts to the chicken. Add lemon juice and cook for 10 minutes until the chicken has cooked through. Garnish with lemon slices, if desired. Enjoy!