Melt 3 tablespoons butter in medium skillet over medium heat. Add shallots, sprinkle with salt and pepper. Sauté until soft and golden, about 10 minutes. Add vinegar and sugar. Stir until brown and glazed, about 3 minutes.
Thinly slice the Brussels sprouts with a mandoline (1/8 inch). Heat oil in large skillet over medium-high heat. Add sprout sprinkle with salt and pepper. Sauté until brown at edges, 6 minutes. Add 1 cup water and 3 tablespoons butter.
Sauté until most of water evaporated and sprouts are tender but still bright green, 3 minutes. Add the shallots, season with salt and pepper.