As far as veggie dishes go, this is tops. Tops, we say.
6 tablespoons (3/4 stick) butter, divided
1/2 pound shallots, thinly sliced
Coarse kosher salt
2 tablespoons apple cider vinegar
4 teaspoons sugar
1 1/2 pounds brussels sprouts, trimmed
3 tablespoons extra-virgin olive oil
1 cup water
Melt 3 tablespoons butter in medium skillet over medium heat. Add shallots; sprinkle with coarse kosher salt and pepper. Sauté until soft and golden, about 10 minutes. Add vinegar and sugar. Stir until brown and glazed, about 3 minutes.
Halve brussels sprouts lengthwise. Cut lengthwise into thin (1/8-inch) slices. Heat oil in large skillet over medium-high heat. Add sprouts; 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 evaporates and sprouts are tender but still bright green, 3 minutes. Add shallots; season with salt and pepper.