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Put a large pot of water (large enough to hold 6 quarts or more of water) with a lid to boil.
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While the pot is heating up, trim the ends and any loose leaves from the brussels sprouts.
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Pierce the ends about ¼ inch deep with a paring knife to help with even cooking.
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When the water is boiling, salt the water heavily.
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Add the prepared brussels sprouts to the water and bring it back to a boil.
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Blanch them for 4-5 minutes, depending on their size.
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Drain the brussels sprouts and rinse with cold water to stop the cooking process. Drain well.
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Cut each brussels sprout in half lengthwise and discard any loose leaves.
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Heat a 10-inch skillet over medium-high heat and add the butter.
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Let the butter completely melt and begin to bubble.
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Add a light amount of salt and freshly ground pepper.
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Add the prepared brussels sprouts and walnuts.
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Toss and stir until the brussels sprouts begin to brown lightly and the walnuts are fragrant.
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Serve immediately.