Heat 1 tablespoon of olive oil in a large (12-inch) saute pan over medium heat. Add the pancetta and cook for 5 to 7 minutes, until browned and cooked through. With a slotted spoon, transfer the pancetta to a plate lined with paper towels and set aside.
Add 2 tablespoons of olive oil and the butter to the same saute pan and cook over medium-high heat just until the butter melts. Add the shallots and cook for 5 to 7 minutes, until tender and starting to brown.
Meanwhile, place the Brussels sprouts in the feed tube of a food processor fitted with the large slicing disk. (You don't need to core them.) Process the sprouts in batches until they are all sliced, transferring them to a bowl. Add the Brussels sprouts to the pan with the shallots, add 1 1/2 teaspoons salt and 3/4 teaspoon pepper and saute for 5 to 7 minutes, stirring occasionally (I use tongs), until crisp-tender and bright green. Add the pancetta and cook for 1 more minute. Off the heat, stir in the balsamic vinegar, season to taste, and serve hot.