Put oven rack in middle position and preheat oven to 500°F.
Heat oil in a 12-inch heavy skillet over moderate heat until hot but not smoking. While oil is heating, sprinkle chicken with 1 teaspoon salt and 1/2 teaspoon pepper, then dredge, 1 piece at a time, in 3/4 cup flour (total), knocking off excess. Brown in 4 batches, turning over occasionally, until golden, 5 to 6 minutes per batch. Transfer to a platter as browned.
Stir together carrots, onions, garlic, cream, wine, lemon juice, mustard, thyme, bay leaf, and remaining teaspoon salt and 1/2 teaspoon pepper in a 5- to 6-quart heavy pot, then bring to a gentle boil over moderate heat, stirring occasionally. Add chicken (chicken will not be completely submerged) and partially cover pot, then braise in oven 25 minutes.
Reduce oven temperature to 350°F and braise 20 minutes more. Transfer chicken to a clean platter and keep warm, loosely covered with foil.
Pour cooking liquid through a sieve into a bowl, discarding solids, then stir in water. Melt butter in a 2 1/2- to 3-quart heavy saucepan over moderately low heat, then whisk in remaining 2 tablespoons flour and cook roux, whisking constantly, 3 minutes. Add cooking liquid in a fast stream, whisking vigorously, then cook at a bare simmer, whisking constantly, 5 minutes.