Preheat the oven to 475°F and line a large baking sheet with parchment paper or lightly grease it.
In a bowl, toss the cubed chicken with the beaten egg and a pinch of black pepper to coat evenly.
Place the flour and ground ginger in a separate shallow bowl and dredge the egg-coated chicken in the flour mixture in batches so pieces are well coated; shake off excess.
Arrange the dredged chicken on one side of the prepared baking sheet and drizzle with 2 tablespoons of the oil.
On the other side of the sheet, toss the broccoli florets, sliced bell peppers, and quartered shallots with the remaining 2 tablespoons oil and spread in a single layer.
Bake the sheet pan at 475°F for 15 minutes.
While the chicken and vegetables start roasting, reduce the oven temperature to 400°F.
Combine the sweet Thai chili sauce, soy sauce, apple juice, rice vinegar, ketchup, peanut butter, sliced jalapeño, and grated garlic in a medium saucepan. Bring to a boil over medium-high heat, then boil 5–8 minutes until the sauce thickens and reduces by about one-third. Remove from the heat.
Pour half of the finished sauce over the chicken on the sheet pan and toss gently to coat; toss the vegetables to distribute any oil.
Return the pan to the oven and bake for an additional 3 minutes, watching closely so the sauce does not burn; the sauce should coat the chicken and be sticky.
Serve the sauced chicken over bowls of rice with remaining sauce on the side and cilantro sprinkled on top; serve the roasted vegetables alongside.