LEMON AIOLI
1 large egg yolk
1 garlic clove, finely grated
1tsp (or more) fresh lemon juice
1/4 cup vegetable oil
Kosher salt
MUSSELS
2 tbsp olive oil
1 medium yellow onion, chopped
Kosher salt, freshly ground pepper
2 garlic cloves, finely chopped
2 tbsp tomato paste
1/2 cup white wine
4 lb mussels, debearder, scrubbed
2 tsp fresh thyme leaves
Sliced country style bread, toasted (for serving)
LEMON AIOLI Whisk egg yolk, garlic, and lemon juice in a medium bowl. Whisking constantly, drizzle in vegetable oil, then olive oil in a slow, steady stream; whisk until aioli is emulsified. Season with salt and more lemon juice, if desired. Cover; chill.
DO AHEAD:Aioli can be made 1 day ahead. Keep chilled.
MUSSELS Heat oil in a large heavy pot over medium- high heat. Add onion, season with salt and pepper, and cook, stirring often, until softened, about 5 minutes. Add garlic and cook, stirring often, until fragrant, about 1 minute. Add tomato paste and cook, stirring until it begins to darken, about 2 minutes. Add wine and bring to a boil. Reduce heat and simmer until liquid is slightly reduced, about 1 minute.
Add mussels and 1/2 cup water to pot, cover, and reduce heat to medium. Cook, stirring occasionally, until mussels open (discard any that do not open), 10-12 minutes.
Ladle mussels and broth into shallow bowls and top with thyme; serve with bread and lemon aioli.