How to Make It
In a large Dutch oven or enameled cast-iron casserole, cover the chicken breasts with the broth, add the sliced carrot and bring to a boil over high heat. Reduce the heat to moderately low, cover and simmer the chicken breasts until just cooked through, about 15 minutes. Transfer the chicken to a plate to cool. Increase the heat to high and continue to cook the carrot until tender, about 10 minutes.
Using a slotted spoon, transfer the carrot to a food processor. Add the tuna, anchovies and mayonnaise and process until smooth. Add the sour cream and 1/4 cup of the chicken broth, season with salt and pepper and blend the tonnato sauce until very smooth.
Slice the chicken breasts on the diagonal 1/4 inch thick. Pour about 1/2 cup of the sauce onto a large serving platter. Top with the arugula and sliced chicken. Pour the remaining sauce over the sliced chicken and garnish with the grape tomatoes, Gaeta olives and capers. Serve the chicken tonnato with lemon wedges and crusty bread.