Put the dried porcini in a bowl, cover with water, and soak for at least 15 minutes.
Cut the chicken into 12 pieces as described. Put the olive oil in a sauté pan large enough to accommodate the chicken snugly and place over medium-high heat. When the oil is hot, 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.
While the chicken is browning, slice the onion. When all the chicken is done, lower the heat to medium-low and add the onion to the pan. Sauté, stirring occasionally, until the onions soften and turns a rich golden color, 5 to 10 minutes.
While the onion is sautéing, thinly slice the fresh mushrooms.
Lift the porcini out of the water and squeeze the excess back into the bowl; do not discard the water. Rinse the mushrooms under running water, then coarsely chop them.
When the onion is done, add the porcini and fresh mushrooms to the pan. Season with salt and cook, stirring occasionally, until the mushrooms are tender, about 10 minutes.
Add the porcini water. Be aware that there may be some sand at the bottom of the bowl, so pour carefully or strain through a paper towel. When the liquid in the pan starts bubbling, adjust the heat so that it simmers and put all the chicken pieces in 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.