In a large pot of boiling water, cook the tortellini until it's just done, according to the package directions. Drain and toss with pesto.
In a large, heavy skillet set over medium-high heat, heat a drizzle of oil and squeeze the sausage out of its casing into the pan. Sauté, breaking the meat up with a spoon, until it's about halfway cooked; add the mushrooms and garlic and cook until the meat is no longer pink and any excess moisture has cooked off. Add the stock or water and stir to scrape the brown bits off the bottom of the pan, then stir in the cream. Cook for a few minutes, stirring, until the sauce thickens slightly.
Add the tortellini to the pan and tear in the spinach. Cook, stirring, for another minute or two, until everything is heated through and the spinach wilts. Serve in shallow bowls topped with Parmesan cheese.