1 large egg yolk
1 garlic clove, finely grated
1 teaspoon (or more) fresh lemon juice
1/4 cup vegetable oil
1/4 cup extra-virgin olive oil
2 tablespoons olive oil
1 medium yellow onion, chopped
Kosher salt, freshly ground pepper
2 garlic cloves, finely chopped
2 tablespoons tomato paste
1/2 cup white wine
4 pounds mussels, debearded, scrubbed
2 teaspoons fresh thyme leaves
Sliced country-style bread, toasted (for serving)
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.
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.