Pat the duck breast dry and score the skin. Season with salt and pepper.
Heat a large nonstick pan over medium heat. Place the duck breast skin side down and render the fat until crispy (6–10 minutes). Remove from pan and let rest for 5 minutes.
Soak the Pemento do Couto in the red wine vinegar with star anise.
Meanwhile toss the broccoli with olive oil, smoked paprika, salt, and pepper. Char it in a pan or over a grill for 3–4 minutes.