Put 2 tablespoons of the sherry vinegar on a rimmed plate. Put the chicken on top of the vinegar, turning it to coat, and allow the chicken to marinate for about 15 minutes at room temperature.

Heat the oil and butter in a large sauté pan over medium-high heat until very hot. Pat the chicken dry and season it well with salt and pepper. Brown the chicken well on both sides. Remove the chicken from the pan and reserve it on a rimmed plate. Add the onion to the pan and lower the heat to medium-low. Allow the onion to cook slowly for about 10 minutes, stirring occasionally. Add the red bell peppers, garlic, and cayenne. Cover the pan and lower the heat to low. Cook until the peppers are very tender, another 15 to 20 minutes. Add the chicken pieces, tomatoes, ham, and remaining sherry vinegar to the pan. Cover the pan and continue to cook until the chicken is cooked through, another 5 to 7 minutes. There should be a very flavorful juice in the pan. Add the chopped watercress to the pan and let it melt into the sauce. Give everything a stir and serve the chicken with the sauce.