In a small bowl, pour 1 cup boiling water over dried porcini. Set aside.
Cut off stems of button mushrooms. Finely chop stems; set aside. Halve caps, slice, and add to stems. With a slotted spoon, lift out porcini, pressing excess liquid into bowl, and transfer to a cutting board. Finely chop porcini and add to stems and caps. Reserve soaking liquid.
Heat olive oil in a large pot over medium-high heat and add onion and salt. Cook, stirring, until onions are soft, about 3 minutes. Add pancetta and cook until onions look a bit creamy, about 2 minutes. Add paprika and cook until very fragrant, 2 minutes. Turn heat to high and add mushrooms. Cook, stirring constantly, until mushrooms start giving off their liquid, 3 to 5 minutes.
Add wine and cook until liquid is reduced by half, about 3 minutes. Add reserved porcini soaking liquid (pouring carefully to leave behind the sandy dregs), chicken broth, 2 cups water, and potatoes. Bring to a boil, then reduce heat to low and simmer, uncovered, until potatoes are tender, 10 to 15 minutes.
Season with salt and pepper to taste. Serve hot, with a dollop of crème fraîche.
Note: Nutritional analysis is per serving.