Position a rack 6 inches from the broiler element and heat the broiler on high.

Line a heavy-duty rimmed baking sheet with foil. Put the tomatoes cut side up on one side and the peppers and onion on the other side. Drizzle everything with 3 tbsp of the olive oil and sprinkle with the cayenne pepper, 1 tsp salt, and 1/4 tsp pepper. Toss the peppers and onions. Broil until the tomatoes are collapsed, about 7 minutes. Flip the tomatoes, toss the peppers and onions, and broil until the tomato skins have large black spots and the peppers and onions are tender, about 5 minutes more.

Use tongs to pull the skins off the tomatoes. With a slotted spoon, transfer the tomatoes to a cutting board. Put the peppers and onions in a large bowl; add the garlic paste. Chop the tomatoes and add to the bowl with the other vegetables. Mix well, season to taste with salt and pepper, and keep warm.

Heat 1/4 cup of the olive oil in a 12-inch non-stick skillet over medium-high heat. Put the flour in a shallow pan. Season the chicken with 1-1/2 tsp salt and 1/2 tsp pepper; dredge in the flour. Working in 2 batches, cook the chicken, flipping once, until just cooked through, 2 to 3 minutes per side. Transfer the chicken to serving plates. Wipe out the pan. Heat the remaining 1 tbsp olive oil and fry the capers over medium-high heat until they pop open and become crisp, about 2 minutes. Sprinkle them over the chicken. Serve with the ragout.