Preheat your oven to 400°F (200°C) and line a large rimmed baking sheet with parchment paper.
Place the sourdough slice into a food processor and pulse until it transforms into coarse, rustic crumbs.
In a shallow bowl, whisk the egg until smooth; in a separate shallow dish, combine the sourdough crumbs with the garlic powder and dried oregano.
Season the chicken breast with half of the sea salt and black pepper, then dip it into the egg wash followed by the sourdough mixture, pressing firmly to ensure an even coating.
Arrange the breaded chicken on one side of the prepared baking sheet.
In a medium bowl, toss the broccoli florets and sliced carrots with the olive oil and the remaining sea salt and pepper until well coated.
Spread the vegetables in a single layer on the other side of the baking sheet.
Roast for 20 to 25 minutes, or until the chicken is cooked through to an internal temperature of 165°F and the sourdough crust is beautifully golden and crisp.