Lemon aioli:

• 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
• Kosher salt

Mussels:

• 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)

For 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.

For 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.