Place the Brussels sprouts in the feed tube of the food processor fitted with a large slicing disc. (There is no need to core them.) Process the Brussels sprouts until they're all sliced.
Place the butter and olive oil in a large (12-inch) saute pan and heat it over medium heat. Add the sliced Brussels sprouts, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper and saute them, stirring frequently, for about 4 to 5 minutes, until tender but still bright green. Stir in the balsamic vinegar, season to taste, and serve hot.
Because you're shredding the Brussels sprouts, there's no need to core them. Place the shredded Brussels sprouts in a plastic bag with a damp paper towel for up to a day.