For filling, in a large skillet cook onion and garlic in hot oil over medium heat about 5 minutes or until onion is tender. Add mushrooms. Cook and stir about 5 minutes more or until mushrooms are tender. Remove from heat. Stir in cheese, spinach, and nutmeg. Set aside.
Using a sharp knife, cut a 2-inch pocket into the thickest part of each chicken breast half by cutting horizontally toward, but not through, the opposite side. Divide filling evenly among pockets in chicken. If necessary, secure openings with wooden toothpicks.
For a charcoal grill, grill chicken on the rack of an uncovered grill directly over medium coals for 12 to 15 minutes or until chicken is no longer pink (170°F), turning once halfway through grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place chicken on grill rack over heat. Cover and grill as directed.) Remove and discard any toothpicks.