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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 1/2 the chicken at a time, skin-side down, for 5 minutes, then turn the chicken and brown 3 to 4 minutes more.
Remove the browned chicken to a plate and add to the pan the sliced fresh mushrooms, browning the mushrooms for 10 to 15 minutes. Add the onion, carrot, celery, garlic, thyme, and bay leaves, salt, and pepper and stir 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 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 325 degree F oven, and heat through 20 to 30 minutes. Serve with quick-cooking polenta or crusty bread, and garnish with chopped celery tops.