Preheat the oven to 475 degrees. Grease baking sheet with oil.
Beat egg in a medium sized bowl. Add salt, pepper, and chicken, tossing to combine.
Add flour to another bowl. Coat egg-soaked chicken in flour, tossing to get a good coat. Lay coated chicken on one Half of the baking sheet and coat with a drizzle of oil. Add halved Brussels sprouts to the other side of the pan, cut side down, and drizzle with oil, tossing with salt and pepper.
Bake for 12 minutes, flip the chicken, toss the brussels sprouts, and cook for another 5 minutes. Then reduce the oven to 400 degrees.
In a medium saucepan combine all the sauce ingredients. Boil for about 6 minutes, stirring occasionally and careful not to burn. Take off heat once the sauce is thickened to run slowly off the back of a spoon.
Pour half the sauce over the chicken and toss the sprouts with a splash of olive oil. Cook for about 3 more minutes.