Core, seed, and peel the pepper. Cut away any white pith inside the pepper and cut it into 1 x ½-inch strips. Peel the eggplant and cut it into ½-inch dice. Peel the tomatoes and coarsely chop. Peel the onions; if using cipolline, quarter them.
Peel and lightly crush the garlic cloves. Put them in a 12-inch skillet with the olive oil and place over medium-high heat. When the garlic has turned a golden brown, remove and discard it. Pat the chicken dry with paper towels and put half of the chicken pieces in the pan. Brown them lightly on both sides, then set them aside on a platter. Repeat with the remaining chicken pieces. Once all the chicken is done, season with salt and pepper.
Put the yellow pepper, eggplant, and onions in the pan and sauté for a couple of minutes. Chop enough of the rosemary leaves to measure ½teaspoon and add it to the pan. Add the tomatoes and season lightly with salt. When the tomatoes start bubbling, adjust the heat so that the tomatoes simmer. Add all the chicken pieces except for the breasts. Cover the pan with the lid slightly askew and cook for 20 minutes, turning the chicken once during that time.
Put the breast pieces in the pan and cook, turning the chicken occasionally, until the dark meat is tender and begins to fall off the bone, 20 to 25 minutes more. If all of the liquid in the pan evaporates before the chicken is done, add a little water. If the chicken is almost done and there is still a lot of liquid in the pan, raise the heat and finish cooking uncovered to allow the excess liquid to evaporate. Serve hot.
This dish will keep well for a couple of days in the refrigerator. Add a couple tablespoons water when reheating it.