Preheat your oven to 425°F (220°C). Line a sheet pan with parchment paper or a silicone mat.
Pat the chicken breast dry with paper towels. In a small bowl, combine olive oil, minced garlic, juice from half a lemon, salt, black pepper, and dried thyme.
Brush the chicken breast thoroughly with the olive oil mixture, ensuring it is well coated. Place the chicken on one side of the sheet pan.
In a separate bowl, toss the broccoli florets with any remaining olive oil mixture from the bowl until evenly covered.
Arrange the broccoli on the sheet pan next to the chicken, ensuring the pieces are spread out to allow for crispy edges.
Roast in the preheated oven for 20-25 minutes, or until the chicken is cooked through (internal temperature of 165°F) and the broccoli edges are crispy. Flip the broccoli halfway through cooking if desired.
Once done, remove from the oven, squeeze a little extra lemon juice over the chicken if desired, and serve immediately.