In a mortar, combine the pimentón de la Vera, cumin, caraway, chopped chipotle in adobo and garlic cloves and pound until a paste forms. Add the olive oil, season with salt and pepper and mix until combined (alternatively, you can use a small blender instead of the mortar and pestle).
Cut the chicken breast into 3 pieces (I like to cut the pointy end and then slice the thicker part transversally). Save 1 Tbs of the paste and spread the rest all over the chicken. Cover and refrigerate for at least 2 hours.
Roast the pepper on a gas stove or broil it in the oven, rotating it so it blackens on all sides. When cool enough to handle, peel it and slice it into thin strips, then into 1-inch pieces, reserving the juices (altenatively, you can use jarred roasted red pepper).
Cut the tomatos: some of them in half transversally, some in half crosswise, some in rings.
Cut the onion into very thin slices.
In a small bowl mix the olive oil with the reserved 1 Tbs pimentón and cumin sauce, chopped cilantro and salt and pepper to taste.
In a medium bowl, add the tomatoes, onion slices and red pepper pieces. Add the dressing and gently stir to incorporate the ingredients.
Brush the chicken with olive oil and grill over moderate to high heat on both sides, about 6 minutes total. Don’t overcrowd the chicken in the pan; if necessary, work in batches.
To serve, arrange the chicken on a platter, then arrange the tomato and pepper salad and the fresh arugula liberally throughout the platter.