Soak the porcini in 1-1/2 cups boiling water for 20 min. Strain them (reserving the soaking liquid) and chop them.
Heat 2 Tbs. of the oil in a large Dutch oven or heavy soup pot over medium heat. Add the pancetta or bacon and cook, stirring occasionally, until crisp, about 7 min.; transfer to a plate lined with paper towels. Season the chicken well with salt and pepper. Add half of the thighs to the pan and brown them well, 2 to 3 min. per side. Transfer to a large plate. Brown the remaining thighs and reserve with the rest.
Add the onion and half of the garlic to the pan and season well with salt. Cook, stirring with a wooden spoon and scraping up any browned bits, until the onion softens and darkens slightly, about 7 min. Add the chopped porcini and cook for another 1 minute. Pour in the sherry and cook, stirring occasionally, until it's almost completely reduced, about 2 min. Add the potatoes, thyme sprigs, chicken broth, reserved porcini water, and vinegar and bring to a boil over high heat. Adjust the heat to maintain a gentle simmer, cover the pan, and cook until the potatoes are just tender, about 15 min. Discard the thyme sprigs. Purée 1-1/2 cups of the chunky potato mixture in a blender. Return the puréed mixture to the rest of the broth, nestle in the chicken (and add any accumulated juices), and cover the pot with the lid slightly ajar. Stew the chicken thighs at a bare simmer, turning them occasionally, until they're cooked through (check by slicing through the bottom of one of the thighs to the bone), about 30 min.
Meanwhile, in a large nonstick skillet, heat the remaining 2 Tbs. oil over medium heat. Add the remaining garlic, sauté until it starts to color, about 20 seconds, and then add the breadcrumbs. Cook, stirring often, until they turn golden brown, about 6 minutes. Stir in 1/2 tsp. of the chopped thyme, season lightly with salt and pepper, and transfer to a plate lined with paper towels.
When the chicken is done, fold in the spinach and stir until wilted, about 1 minute. Stir in the remaining 1 tsp. chopped thyme and the pancetta; season well with salt and pepper. Serve immediately, sprinkled generously with the breadcrumbs.