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Preheat oven to 375° F.
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Lightly coat 9 x 9-inch baking pan (or casserole dish) with spray. Set aside.
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Combine Neufchatel cheese and 1 tsp. garlic powder in a small bowl; mix well.
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Place slices of ham individually on your work surface. Top each with a slice of Swiss cheese. Spread Neufchatel cheese mixture evenly on each slice of Swiss cheese. Roll lengthwise. Place in freezer until ready to use.
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Combine bread crumbs, salt, remaining ¼ tsp. garlic powder, pepper, and onion powder in a small bowl; mix well. Transfer to a plate. Set aside.
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To stuff chicken, slice chicken breasts along the long side, about ¾ way through; do not cut all the way through. Chicken will open like a book. Place one ham roll inside each chicken breast. Secure with a toothpick. Repeat with remaining chicken breasts. Set aside.
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Place eggs in a shallow dish. Dip each chicken breast in eggs, completely coating, then bread crumbs; coating each evenly on all sides. Place chicken in prepared pan. Remove toothpicks.
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Bake for 25 to 30 minutes, or until chicken is no longer pink in the middle and juices run clear.