Heat ¼ inch oil in a large skillet over medium to medium-high heat until shimmering. Dredge the chicken in the egg mixture then the cornstarch and place in the oil. If the first piece of chicken doesn't sizzle, wait 2 more minutes before adding the rest. Fry the chicken for about 3 minutes per side until golden, flipping once. Work in batches if you need to to avoid crowing the pan. Remove the chicken to a plate lined with paper towels while you make the rest of the sauce.

When the chicken is cooked, drain all but 1 tablespoon oil from the skillet, and reduce the heat to medium. Add the broccoli and toss for 2 minutes, then add ¼ cup water to help the broccoli steam. Steam the broccoli for 2 minutes. When bright green but still very crunchy, add the Campbell's sauce, rice vinegar, and sesame oil. Taste, and add sugar if desired. Stir to coat the broccoli, then add the chicken back to the mixture for 1 minute to heat through. Serve topped with the scallions over rice, if desired.