In a food processor, whirl egg yolks, mustard, salt, pepper, and 1 garlic clove until smooth. With motor running, add olive oil in a thin, steady stream until mixture is emulsified. Add lemon juice and pulse to blend. Transfer aioli to a bowl, cover, and chill until ready to serve.
In a 4-qt. saucepan, combine wine, remaining 3 garlic cloves, and onion. Bring to a boil over high heat, then reduce heat and simmer 5 minutes. Add mussels, cover, and simmer until shells have opened, about 6 minutes.
With a slotted spoon, scoop mussels from pan into a serving bowl, discarding any that haven't opened. Whisk butter into wine mixture until melted, then stir in parsley. Pour sauce over mussels and serve immediately with the aioli.
Wine note: Pinot Gris/Grigio, the West is awash in wine from this grape; it's mostly called Pinot Grigio and made in the steely, citrusy Italian style. The French--and Oregonians--call it Pinot Gris and give it rounded layers of melon and pear with tangy, earthy minerals alongside the citrus. Pinot Gris works best with fairly simple seafood and poultry, so the layers don't get muddled, but it gets along fine with a lot of herbs. We also like it with lemony sauces such as aioli.
Claiborne & Churchill Pinot Gris 2005 (Arroyo Grande; $18). Creamy stone fruit sweetens the mussels.
Rex Hill Vineyards Pinot Gris 2005 (Oregon; $17). Lean and minerally, with tart apricot balancing creamy lemon flavors; this wine matches both the sweetness and the garlicky brininess of the mussels.
WillaKenzie Estate Pinot Gris 2005 (Willamette Valley; $18). Citrus, creamy peaches and apricots, flowers, and minerals; it "has its own moment with the mussels," according to one of our tasters.
Note: Nutritional analysis is per serving.