Place the dried mushrooms and stock in small pot over medium-high heat. Bring to low boil, then reduce the heat to low and reconstitute.
Heat a thin layer of olive oil, a couple of turns of the pan, in a large skillet with a tight-fitting lid or a Dutch oven over medium-high heat. Pat the chicken pieces dry and season liberally with salt and pepper. To avoid crowding the pan, brown half the chicken at a time, skin-side down, for 5 minutes, then turn the chicken and brown for 3-4 minutes more.
Remove the browned chicken to a plate and add the sliced fresh mushrooms to the pan, browning them for 10-15 minutes. Add the onions, carrots, celery, garlic, thyme, bay leaves, salt and pepper and stir for 10 minutes more to soften.
Deglaze the pot with the Marsala. Add the chicken pieces back to the pot and arrange the vegetables and mushrooms on top. Remove the porcinis from the small pot; roughly chop and add to the chicken. Pour the porcini stock over the chicken, reserving the last couple of spoonfuls, as this is where any grit may have settled.
Cover and braise at a low simmer for 30 minutes over low heat. Cool and store for a make-ahead meal. Reheat over a medium flame, or in a preheated 325ºF oven, and heat through, 20-30 minutes.
Serve with quick-cooking polenta or crusty bread and garnish with chopped celery tops.