Season chicken pieces with salt and pepper. Dredge in flour, shaking off excess.
In a large Dutch oven or heavy pot, melt 2 tablespoons butter over medium-high heat. Brown chicken in batches, skin-side down first, until golden on all sides, about 8 minutes per batch. Transfer to a plate.
Reduce heat to medium. Add remaining 2 tablespoons butter. Add pearl onions and mushrooms; cook, stirring occasionally, until lightly browned, about 5 minutes. Add garlic and cook 1 minute.
Pour in white wine, scraping up any browned bits. Add chicken broth and bouquet garni. Return chicken to pot, skin side up. Bring to a simmer.
Cover and cook over low heat until chicken is tender and cooked through, about 45 minutes. Remove bouquet garni.
Stir in heavy cream and simmer uncovered until sauce thickens slightly, about 5 minutes. Adjust seasoning with salt and pepper.
Sprinkle with fresh parsley and serve.
Tips
For a richer flavor, marinate the chicken in the white wine overnight in the refrigerator.
Leftovers can be stored in an airtight container in the refrigerator for up to 3 days.
Reheat gently on the stovetop, adding a splash of broth or cream if needed.