Shred 1 pound Brussels sprouts in a food processor. Saute 4 thinly sliced shallots in 2 tablespoons butter over medium heat, 3 minutes. Season with salt and pepper. Add 2 tablespoons cider vinegar and 1 tablespoon sugar and cook, stirring, 2 minutes. Transfer to a plate and wipe out the skillet. Add 1 tablespoon butter to the skillet and increase the heat to medium high. Add the Brussels sprouts, and salt and pepper to taste; cook 5 minutes. Add 1/2 cup water and a pat of butter; cook 5 more minutes. Stir in the shallots.