Preheat your oven to 425°F.
Prepare a baking sheet by lightly coating it with olive oil or non-stick spray.
Pat the chicken breast dry and place it on a cutting board. If needed, slice it horizontally to form thinner cutlets for quicker and more even cooking.
In a shallow bowl, whisk the egg until smooth. In another shallow bowl, spread out the crushed everything bagel crumbs.
Dip each chicken piece first in the egg, allowing any excess to drip off, then coat thoroughly in the bagel crumbs, pressing gently to help the crumbs adhere.
Arrange the breaded chicken on the prepared baking sheet.
On another baking tray, toss the broccoli florets with a drizzle of olive oil, salt, and pepper to taste.
Place both trays in the oven. Roast the broccoli for about 15 minutes, stirring halfway through.
Bake the chicken for approximately 18-20 minutes or until it reaches an internal temperature of 165°F and the coating is crispy. Optionally, finish the chicken under the broiler for 2 minutes for extra crispiness.
Remove both trays from the oven. Plate a portion of roasted broccoli alongside the crispy bagel chicken and serve immediately.