This is really a Swiss recipe (not French, but I chose French cuisine because it was the closest category). The dish is called Poulet a la Fermiere in Switzerland and, as the name suggests, contains a farm's bounty of fresh chicken, delectable cheese, and plenty of fresh vegetables and aromatic herbs.
Heat butter in a 12-inch ovenproof deep heavy sauté pan over moderately high heat until foam subsides, then brown chicken all over, in batches if necessary, 8 to 10 minutes. Transfer to a plate and cover. Pour off all but 1 tablespoon fat from pan.
Tie parsley, thyme, and bay leaf in cheesecloth to make a bouquet garni, then add to pan with carrots and onions, stirring to coat with fat. Add wine and deglaze by boiling over high heat, stirring and scraping up brown bits, until liquid is reduced by half, about 3 minutes. Add broth and chicken, skin sides up, with any juices from plate, and simmer, covered, 10 minutes. Add potatoes and salt and pepper to taste and simmer, covered, until chicken is cooked through and potatoes are tender, about 15 minutes.
Discard bouquet garni. Stir in crème fraîche, peas, and salt and pepper to taste, then turn chicken in sauce to coat. Sprinkle dish all over with Gruyère, and broil 4 to 5 inches from heat until browned and sauce is bubbling, about 3 to 4 minutes.