Prepare a large bowl of ice water. Set aside.
Trim the Brussels sprouts and peel off 1 outer layer of leaves. Cut the sprouts in half lengthwise, larger sprouts can be quartered. Place the sprouts in a large saucepan, cover with cold water and add about 2 tablespoons of salt. Bring the sprouts to a boil and drain immediately. Transfer the sprouts to the bowl of ice water. (This will stop the cooking and preserve their color.) Drain well.
Heat a large skillet over medium-high heat, add the butter. When the butter is melted, add the Brussels sprouts. Cook for 1 minute then add the sliced cabbage. Cook until the cabbage has softened and the sprouts are warmed through. Season with salt and pepper and serve warm.
To easily drain vegetables from an ice bath, place a small colander over a bowl of ice water. Place the cooked vegetables into the colander and dunk into the water. Remove the colander and drain the veggies, leaving the ice and water behind.