Place the Brussels sprouts in the feed tube of the food processor fitted with a large slicing disk. (You don't need to core them.) Process until they're all sliced.
Heat the butter and olive oil in a very large (12- to 14-inch) sauté pan over medium to medium-high heat. Add the Brussels sprouts, 1 1/2 teaspoons salt and 3/4 teaspoon pepper and saute, stirring occasionally, for 5 to 7 minutes, until crisp-tender and still bright green. Off the heat, stir in the balsamic vinegar, season to taste and serve hot.
Make ahead: Prep the Brussels sprouts and place in a plastic bag with a damp paper towel for up to a day. Saute before serving.