Preheat oven to 425 degrees F. Line a baking sheet with parchment or foil.
Toss halved Brussels sprouts with olive oil, salt, pepper, and balsamic or honey if using. Arrange cut-side down on one half of the baking sheet.
Pat chicken dry. In a small bowl, mix sour cream with garlic powder, onion powder, salt, and pepper. Spread evenly over both sides of each breast.
In a shallow bowl, combine Parmesan, panko, smoked paprika, Italian seasoning, and olive oil. Press each coated breast firmly into the breadcrumb mixture to coat both sides.
Place breaded chicken on the other half of the baking sheet.
Roast 20 to 25 minutes, flipping Brussels sprouts halfway through, until chicken is cooked through and breading is golden and crisp.
For extra crispiness, broil 1 to 2 minutes at the end. Watch closely.