Cut chicken into 1-1/2-inch pieces. In a shallow dish combine flour, sesame seeds, pepper, and salt. Place eggs in another shallow dish. Dip chicken pieces into flour mixture, then into eggs, then into flour mixture again to coat. Set aside.
In a 12-inch skillet heat about 1/2 inch oil over medium-high heat. Add chicken pieces, half at a time, and cook for 4 to 5 minutes or until golden brown and cooked through (170 degrees F). Using a slotted spoon, transfer to paper towels to drain.
Meanwhile, in a large skillet combine carrot, honey, vinegar, and the water. Bring to boiling. Stir in green beans. Add chicken; stir to coat. If desired, serve over hot cooked rice.