Cut off base of a sprout with a sharp knife. Slice a piece from one side of the sprout. Place it cut side down and cut it crosswise into about 5 slices. Repeat with all of the sprouts. (I sliced them all).
Heat a large heavy skillet over medium heat for 1 minutes. Add olive oil and swirl to coat skillet.
Add shallots; sauté for 2 mintues. Stir in sprouts, salt (if using), and 3 tablespoons water; spread everything evenly across the pan. Reduce heat to medium-low, cover, and cook for 5 minutes.
Meanwhile, combine mustard, brown sugar, and remaining water in a small bowl; stir until blended. Once sprouts have cooked for 5 minutes, pour mustard mixture over them and stir. Reduce heat to low, cover, cook for 3 minutes, then stir. Remove from heat or, if you prefer, cook for 5 minutes more – the sprouts will get browner and more intensely flavored. Serve hot or warm with black pepper.