Preheat the oven to 475°F and line a baking sheet with parchment or lightly grease it.
In a bowl, combine the cubed chicken with the beaten egg and a pinch of black pepper; toss to coat.
Place the flour in a separate shallow bowl. Working in batches, dredge the egg-coated chicken in the flour until lightly coated and arrange on one side of the prepared baking sheet; drizzle with 2 tablespoons of oil.
On the other side of the baking sheet, toss the broccoli florets and sliced shallots with 2 tablespoons of oil and salt and pepper to taste.
Bake for 12 minutes, then toss the broccoli, flip the chicken pieces, and bake 3–5 more minutes until the chicken is cooked through and the vegetables are tender.
While the chicken bakes, combine pomegranate juice, soy sauce, honey, rice vinegar, grated garlic, grated ginger, star anise, bay leaf, and crushed red pepper flakes in a large skillet; bring to a boil over medium-high heat.
Boil the sauce 5–8 minutes until it thickens and reduces by about one-third.
Add the roasted chicken, broccoli, and toasted sesame seeds to the skillet and cook, stirring, until the sauce evenly coats the chicken and vegetables and is glossy.
Serve the sauced chicken and broccoli over ginger rice and top with sliced green onions.