How to Make It
Preheat the oven to 425°. Season the chicken with salt and pepper. In a large ovenproof skillet, heat 1 tablespoon of the olive oil over moderately high heat. Add the chicken skin side down and cook until very lightly browned, about 3 minutes. Turn the chicken skin side up and transfer the skillet to the oven; roast the chicken for about 10 minutes, until it is cooked through. Transfer the roasted chicken to a work surface and let cool.
Meanwhile, bring a medium saucepan of salted water to a boil. Add the potatoes and cook until tender, about 10 minutes. Drain the potatoes and let cool.
In a blender, combine the piquillo peppers with the garlic, vinegar, honey and lemon juice and puree until very smooth. With the machine on, add the remaining 1/4 cup of olive oil in a thin stream and blend until the dressing is emulsified. Season the dressing with salt and pepper.
Discard the chicken skin and thinly slice the breasts crosswise. In a large bowl, toss the chicken with the potatoes, olives and carrot and toss with the dressing. Add the mesclun, toss gently and serve.