Cut the stalks off the fennel. If there are any fronds, save a small handful. Quarter the fennel bulb through its core; reserve one quarter (two if using small bulbs). Slice the remaining fennel through the core into wedges about 1/2 inch thick.
Season the chicken well with salt and pepper. Heat 3 tablespoons of the oil in a large Dutch oven or heavy soup pot over medium-high heat. Add half of the thighs and brown them well, 2 to 3 min. per side. Transfer to a large plate. Add the remaining thighs and the sausage to the pot, brown them, and reserve with the first batch of thighs.
Reduce the heat to medium, add the fennel wedges and onion, and season well with salt. Cook, stirring often until the onion softens completely and caramelizes slightly, about 10 min. Push the fennel and onion to one side of the pot, add 1 tsp. of oil to the pot and sauté the garlic in the oil for 20 seconds. Stir it into the vegetable mixture, turn the heat to high, and add the wine. Cook, scraping the bottom of the pan, until the wine is almost completely reduced, about 2 min. Add 1-1/2 cups of the chicken broth, the potato, and the rosemary. When the mixture reaches a boil, reduce the heat to medium low and cover the pot. Simmer until the potato is just tender, about 15 min. Transfer 1-1/2 cups of the vegetables and broth to a blender, purée, and then mix it back into the rest of the broth. If the sauce is too thick, thin it with some of the remaining 1 cup broth. Nestle the chicken into the sauce, add any accumulated juices, cover the pot with the lid slightly ajar, and adjust the heat to maintain a low simmer. Slice the sausages into 1-inch pieces. After the chicken has been stewing for 15 min., add the sausage pieces. Cover partially and simmer gently, stirring occasionally, until the chicken and sausage are cooked through (check by slicing into each), another 15 to 20 min. Discard the rosemary.
Meanwhile, use a sharp vegetable peeler to shave the remaining fennel and put it in a small bowl. Season well with salt and pepper. Finely chop 1 to 2 Tbs. of the reserved fennel fronds (save the rest) and add to the shaved fennel along with the remaining 1 teaspoon oil. Toss well and taste for seasoning.
Use a sharp peeler to shave thin slices of fennel. Start peeling on the inside of the bulb, using long, smooth strokes.
Stir the balsamic vinegar and Parmigiano into the stew; taste and adjust the seasonings. Serve, ladling a portion of chicken, sausage, and sauce into each bowl. Sprinkle each serving with more Parmigiano and mound a small portion of fennel salad and the reserved fennel fronds over it.