Preheat the oven to 450 degrees F. Cook the rice as the label directs. Transfer to a medium bowl. Toast the sesame seeds in a medium nonstick skillet over medium-high heat, stirring occasionally, until browned, 3 to 5 minutes. Add to the rice.
Return the skillet to medium-high heat. Add the vegetable oil, then add the shallot, water chestnuts and ginger. Cook, stirring, until the shallot is tender and lightly browned, about 3 minutes. Add to the rice.
Mix 1 tablespoon soy sauce with 1 teaspoon honey until smooth. Stir into the rice along with half the scallions; season lightly with salt and pepper.
Halve each chicken breast horizontally, cutting three-quarters of the way through. Open the chicken like a book; season with salt and pepper. Divide the rice among the chicken, firmly pressing the mixture together on one side of each breast. Fold the chicken halves back together and secure with a few toothpicks. Rub with the remaining 1 teaspoon each soy sauce and honey; season with salt and pepper.
Transfer the chicken to a baking sheet. Drizzle with vegetable oil and roast until a thermometer inserted into the stuffing registers 165 degrees F, 20 to 25 minutes. Discard the toothpicks and sprinkle the chicken with the remaining scallions.