1. Remove the loose and faded leaves and trim the stalk ends from the Brussels sprouts.
2. Bring about 1 inch of lightly salted water to a boil in a 2-quart saucepan. Add the Brussels sprouts, and cook, covered, for 8 to 10 minutes, or until they are tender but still bright green. Drain them in a colander and rinse them quickly with cold water to preserve their color and to stop the cooking.
3. Melt the butter in an 8-inch skillet, add the walnuts, and cook, uncovered, over low heat for 3 minutes, or until the nuts brown lightly. Add the Brussels sprouts and the salt and pepper, and continue cooking about 2 minutes—or just long enough to heat through.