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) saute 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.
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