In a small bowl, combine the black pepper, allspice, oregano and 1 teaspoon salt. Sprinkle this mixture evenly over both sides of the chicken breasts.
Heat the oil in a very large (12-inch) skillet over medium. Lay in the chicken (skin-side down if you are using skin-on chicken), and cook until richly browned, about 2 minutes on each side (3 to 4 minutes if using skin-on chicken). Remove the chicken to a plate, leaving behind as much oil as possible.
Add the sliced onion and carrots to the skillet and cook, stirring regularly, until the onions are browned, about 7 minutes. Add the garlic, stir for about 1 minute, then add the vinegar, jalapeños and the broth. Nestle the chicken pieces, into the onion mixture, and simmer gently over medium-low heat until the chicken is just cooked through, about 5 minutes (about 15 minutes for skin-on, bone-in chicken).
Taste the broth, season with additional salt if you think it’s appropriate. Transfer a piece of chicken to each dinner plate, spooning a generous portion of the juicy vegetable mixture over the top. Dinner’s ready.