Trim the ends and outer leaves from the brussels sprouts, then rinse them and pat dry with paper towels.
Process the sprouts using the shredding disk of your food processor or cut them into very thin slices by hand and separate them into shreds. Set aside.
Put a large pot of salted water on to boil for the pasta.
Fry the bacon until crisp, then transfer to a paper towel-lined plate. Drain all but 2 tablespoons of the bacon fat from the pan.
Put the pan back on the stove over medium heat and add 2 tablespoons of olive oil. Add the onion and sauté until it's soft and translucent, 4 to 6 minutes. Add the garlic and cook for 1 minute longer.
Start adding the Brussels sprouts, a handful a time, letting each addition wilt slightly before adding more. Season with salt and pepper and continue to cook, stirring often, until the Brussels sprouts are tender and just beginning to caramelize, 5 to 7 minutes.
Drizzle the sprouts with the vinegar and add the reserved bacon and a bit more olive oil if desired. Taste and adjust the seasoning if needed, then reduce the heat to low to keep warm.
Cook and drain the pasta according to package directions, reserving 1/4 cup of the cooking liquid.
Return the pasta to the pot, add the Brussels sprouts and toss to combine. Stir in 2 to 3 tablespoons of the cooking liquid, then add the cheese and combine once again.
Serve immediately and pass extra cheese at the table if desired.