To clarify butter: Melt ½ stick butter over low heat. When melted, remove from heat and set aside for several minutes to allow milk solids to settle. Skim clear butter from top and discard sediment.
Make lemon butter sauce: Heat clarified butter in a small pan, add finely chopped onion and garlic, and sauté until transparent. Add lemon juice and white wine, season with salt and pepper. Simmer 2-3 minutes to reduce liquid. Remove from heat and swirl in cold butter until sauce is smooth and emulsified.
Prepare mussels: Soak mussels in cold water for several minutes. Scrub with a stiff brush and remove beards with a sharp knife or by pulling with a damp cloth. Rinse again in cold water.
Heat olive oil in a 10-inch skillet and add mussels. Cover with another skillet or lid and cook until shells begin to open, about 2 minutes.
Remove top, add chopped onion and garlic, and toss. Cover again and cook for 1 minute.
Remove top and add Pernod, basil, lemon juice, and lemon butter sauce. Return to heat for 30-45 seconds with top off skillet.
Discard any mussels that did not open. Serve immediately in deep bowls.