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Right before cooking, scrub mussels with a stiff sponge or brush under cold running water. (If the "beards" are still attached, trim them with a knife.)
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While cleaning mussels, discard ones with broken shells. Lightly tap the hinge end of any open shells on a work surface. Discard those that don't close.
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In a large pot with a lid, melt 1 tablespoon butter over medium-high. Add shallot and garlic. Cook, stirring, until fragrant, 1 minute.
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Add white wine and chicken broth. Bring to a simmer over high.
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Add mussels; stir to combine. Cover, reduce heat to medium-high, and cook until mussels open, 5 minutes, stirring once. Discard unopened mussels. (It's not unusual to lose a small portion of mussels at this point.)
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Season with salt and pepper. Remove from heat; stir in parsley and remaining butter until butter is melted. Serve with crusty bread.