Heat the oil in a large Dutch oven over medium-high heat until shimmering hot, about 1 1/2 minutes.
Add the sausage and cook, tossing occasionally, until well browned all over, about 5 minutes.
Stir in the fennel, sprinkle lightly with salt and cook, stirring until it browns all over and starts to soften, about 5 minutes.
Add the broth and vinegar and bring to a boil.
Reduce the heat to a simmer, cover with the lid slightly askew, and cook until the sausage cooks through and the fennel is tender, about 15 minutesslice into one of the thicker pieces of each to check.
Stir in the chicken, cream, Tabasco, and thyme and cook another 10 minutes to heat through.
Season with salt and pepper to taste.
Serve immediately with a sprinkling of the Parmesan.