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HEAT oven to 400°F.
CUT cheese crosswise in half, then cut each piece lengthwise into 4 sticks. Wrap 1 chicken breast around each cheese stick to completely enclose cheese; secure with wooden toothpicks. Dip chicken roll-ups, 1 at a time, in eggs then in bread crumbs, turning to evenly coat each roll-up.
HEAT 2 Tbsp. oil in large skillet on medium-high heat. Add chicken; cook 5 min. or until evenly browned, turning occasionally. Transfer to 13x9-inch baking dish sprayed with cooking spray.
BAKE 15 to 18 min. or until chicken is done (165ºF). Meanwhile, wipe skillet with paper towels. Add remaining oil and vegetables to skillet; cook 8 to 10 min. or until crisp-tender, stirring occasionally. Stir in flour and pepper until blended. Gradually stir in broth; cook 3 to 5 min. or until thickened, stirring constantly. Add rice; cook 2 min. or until heated through, stirring occasionally.
DISCARD toothpicks from chicken. Serve chicken over rice mixture.