In a large skillet over medium heat, heat the oil. Add the shallot and sauté until translucent. Stir in the sausage and thyme and continue to cook until the sausage is nicely browned. Remove from the heat and stir in the breadcrumbs and chicken stock. Season to taste with salt and pepper and set aside to cool.
Prepare your barbecue to grilling at medium heat, or preheat your broiler to medium.
With a small, sharp knife, cut a pocket in the thickest part of each chicken breast. Stuff each pocket with the cooled stuffing. Season the chicken breasts with salt and pepper. Grill over hot coals until the center of each breast reads 160º-165º on an instant read thermometer, about 5-8 minutes per side. Allow to rest 5 minutes before serving.