Heat a dry large skillet over medium heat. Add bacon; cook, stirring occasionally, until crisp, 5 to 7 minutes. Using a slotted spoon, transfer to a paper-towel-lined plate to drain. Add enough oil to bacon fat in skillet to total 2 tablespoons. Add garlic, shallots, and sprouts; cook, stirring occasionally, until pale golden, about 3 minutes. Add stock; season with salt and pepper. Cook until most of the liquid has been absorbed and sprouts are tender, 10 to 12 minutes.
Meanwhile, bring a large pot of water to a boil; add 1 tablespoon salt and the pasta. Cook according to package directions until al dente. Drain, reserving 1/2 cup cooking liquid; return pasta to pot.
Stir in sprouts mixture, reserved 1/2 cup cooking liquid, and cheese. Drizzle with 2 tablespoons oil; add sage and bacon. Toss until combined. Serve with more cheese.