Ingredients

• 3 tablespoons unsalted butter
• 2 tablespoons olive oil
• 1 large onion, chopped
• 1 garlic clove, finely chopped
• 6 cups corn kernels (from 6 ears)
• Kosher salt, freshly ground pepper
• 2 sprigs thyme
• Kosher salt
• 2 1 1/4-pound live lobsters
• 1/2 small shallot, finely chopped
• 2 teaspoons fresh lemon juice
• 1 teaspoon Dijon mustard
• 2 tablespoons extra-virgin olive oil plus more for drizzling
• Freshly ground black pepper
• 3 tablespoons coarsely chopped celery leaves plus more for garnish
• 1 tablespoon coarsely chopped fresh tarragon plus leaves for garnish
• Flaky sea salt (such as Maldon)

Instructions

1. Heat butter and oil in a large pot over medium-high heat. Add onion and garlic and cook, stirring often, until softened and beginning to brown, about 5 minutes. Add corn; season with salt and pepper. Cook, stirring occasionally, until corn is softened and beginning to brown, 6-8 minutes.
2. Add thyme sprigs and 5 cups water to pot. Bring corn mixture to a boil, reduce heat, and simmer until corn is very soft, 6-8 minutes longer; discard thyme sprigs.
3. Working in batches, purée corn mixture in a blender until smooth (or use an immersion blender in the pot). Strain through a fine-mesh sieve into a large bowl, pressing on solids; discard solids. Transfer soup to a large bowl, cover, and chill until cold, at least 2 hours.
4. For lobster salad and assembly: Pour water into a large pot to a depth of 1"; bring to a boil and salt generously. Add lobsters, cover, and cook until bright red, 8-10 minutes. Transfer lobsters to a rimmed baking sheet and let cool. Crack lobster shells, pick meat from tail and claws, and cut into bite-size pieces (you should have about 2 cups meat). Whisk shallot, lemon juice, and mustard in a medium bowl. Gradually whisk in 2 tablespoons oil; season dressing with kosher salt and pepper. Add lobster meat, 3 tablespoons celery leaves, and 1 tablespoon tarragon; toss gently to coat. Cover and chill until cold, about 1 hour. Divide soup among bowls and top with lobster salad and tarragon and celery leaves. Drizzle with oil and season with sea salt and pepper. DO AHEAD: Lobster can be cooked and picked 1 day ahead. Cover and chill. Soup and lobster salad can be made 8 hours ahead. Keep chilled. Toss lobster meat with celery leaves, tarragon, and dressing just before serving.
5. Pour water into a large pot to a depth of 1"; bring to a boil and salt generously. Add lobsters, cover, and cook until bright red, 8-10 minutes. Transfer lobsters to a rimmed baking sheet and let cool.
6. Crack lobster shells, pick meat from tail and claws, and cut into bite-size pieces (you should have about 2 cups meat).
7. Whisk shallot, lemon juice, and mustard in a medium bowl. Gradually whisk in 2 tablespoons oil; season dressing with kosher salt and pepper. Add lobster meat, 3 tablespoons celery leaves, and 1 tablespoon tarragon; toss gently to coat. Cover and chill until cold, about 1 hour.
8. Divide soup among bowls and top with lobster salad and tarragon and celery leaves. Drizzle with oil and season with sea salt and pepper.
9. DO AHEAD: Lobster can be cooked and picked 1 day ahead. Cover and chill. Soup and lobster salad can be made 8 hours ahead. Keep chilled. Toss lobster meat with celery leaves, tarragon, and dressing just before serving.