Wash the Brussels sprouts under running water. Slice off the stem end about 1/4 way into the sprout and then remove the yellowing outer leaves. Check to see that the remaining sprout is completely clean and the leaves unblemished, if not wash again and remove another layer of outer leaves. With a sharp knife, cut an X into the stem end. This helps the interior core cook at the same rate as the outer leaves.
In a large frying pan (I like to use my cast-iron skillet), heat the olive oil over medium-high heat almost to the smoking point. Saute the Brussels Sprouts until they just start to brown. Add the balsamic vinegar, turn the heat to low, and cover. After about 2 minutes, remove cover and add 1/4 cup of water. Cover and continue cooking until the sprouts start to soften, approximately 8 to 10 minutes. NOTE: They should be brown but not burned.
Turn the heat up to high, add the butter and chopped pecans. Saute for 1 to 2 minutes or until the pecans are lightly toasted. Remove from heat and add salt and pepper to taste. Serve immediately.
Makes 4 to 6 servings.