• Soak mussels in cold water for several minutes, then scrub with a stiff brush and remove "beard" (the little tuft of fibers protruding from the shell), either with a sharp knife or by pulling on it with a damp cloth. Rinse mussels again in cold water. 
  
• Heat olive oil in a 10-inch skillet; over medium heat add mussels. Cover with another 10-inch skillet or lid and cook until shells begin to open about 2 minutes. Remove top and add onion and garlic and toss. 
  
• Cover pan again and cook for 1 minute. Remove top and add Pernod, basil, lemon juice, and lemon butter sauce. Return to flame for 30 to 45 seconds with the top of the skillet. Discard any mussels that did not open. Serve in a deep bowl. LEMON Simmer 2 to 3 minutes to reduce liquid. Remove from heat and swirl in cold butter until sauce is smooth and emulsified.
  

Butter Sauce Recipe

• BUTTER SAUCE: 2 tablespoons clarified butter (you'll need about 1/2 stick butter; directions follow) 2 tablespoons finely chopped yellow onion 2 tablespoons finely chopped garlic 6 tablespoons fresh lemon juice 2 tablespoons dry white wine Kosher salt White pepper 2 tablespoons cold butter.
• To clarify butter: Melt 1/2 stick butter over low heat. When melted, remove from heat and set aside for several minutes to allow the milk solids to settle to the bottom. Skim the clear (clarified) butter from the top and discard sediment. (This can be done ahead.) 
• To make the sauce: Heat clarified butter, add onion and garlic and saute until transparent. Add lemon juice and white wine and season to taste with salt and pepper.