Put a baking sheet on the top oven rack and preheat to 475 degrees F. Toss the potatoes with 1 tablespoon olive oil in a bowl; season with salt and pepper. Spread out on the hot baking sheet and roast until the potatoes start browning, about 10 minutes.
Meanwhile, heat 1 tablespoon olive oil in a large skillet over medium-high heat. Pat the chicken dry with paper towels and season with salt and pepper. Cook skin-side down until the skin is crisp and brown, about 6 minutes, then flip and brown lightly on the other side, about 3 minutes.
Make the sauce: Blend the chiles, garlic, lemon zest and juice, 1 tablespoon vinegar, the paprika, honey and 1 teaspoon salt in a blender to make a chunky puree. Gradually blend in 1/4 cup olive oil. Set aside half of the sauce for serving.
Move the potatoes to one side of the baking sheet and add the chicken skin-side up to the other side. Brush all over with the remaining chile sauce. Roast until the chicken is cooked through and the potatoes are browned and crisp, 13 to 15 minutes. Divide the chicken, potatoes and pan juices among plates.
Toss the lettuce mix with the remaining 1 tablespoon olive oil and 2 teaspoons vinegar; season with salt and pepper. Add the salad to the plates. Serve with the reserved chile sauce.