In a heavy kettle, heat the oil over moderately high heat until it is hot but not smoking and in it saute the chicken pieces, seasoned with salt and pepper, in batches, turning them once, for 8 to 10 minutes, or until they are browned. Transfer the chicken pieces as they are browned to a plate and keep them warm, covered.
Place the shallot, onions, carrots, pearl onions, bay leaf, and thyme in a slow cooker. Place the browned chicken on top of the vegetables. Combine the Cognac, white wine and 1 3/4 cup of chicken broth and add to the cooker. Cover and cook on low for 5 hours or until chicken is tender.
Turn control to high and add the mushrooms. Dissolve the cornstarch in the remaining 1/4 cup chicken stock. Stir into the mixture. Cover and cook for 20 minutes more, or until sauce is thickened, stirring once. Transfer the chicken to a plate, and keep warm. Discard the bay leaf, and season the sauce with freshly ground nutmeg and salt and pepper to taste. Serve warm