Sweat the veggies. Heat olive oil in a large soup pot or Dutch oven over medium heat. Add diced onion, carrots, and fennel, then cook for 5–7 minutes, stirring occasionally, until softened but not browned.
Build the base. Add diced potato, 2 crushed garlic cloves, and 3 sprigs each of thyme and tarragon (tied together with kitchen twine). Season with 1 tsp Kosher salt and ¼ tsp black pepper. Stir in 2 Tbsp tomato paste, breaking it up and cooking for 1–2 minutes to deepen the flavor.
Deglaze and simmer. Pour in 1 cup Cognac. Either carefully ignite with a long lighter to burn off the alcohol, or let it simmer until reduced and the alcohol has cooked off. Add 3 cups seafood stock, bring to a gentle simmer, and cook for 30 minutes, until the vegetables are very tender.
Blend until silky. Remove and discard the herb bundle. Use an immersion blender to purée the soup until completely smooth (or transfer carefully to a blender). For an ultra-luxurious texture, pass the soup through a fine-mesh sieve to remove any remaining solids.
Finish the bisque. Stir in 1 cup heavy cream, then fold in 1 lb fresh crab meat, reserving a small amount for garnish. Taste and adjust seasoning as needed (you’ll likely add about ½ tsp more Kosher salt, depending on your stock). Squeeze in the juice of ½ lemon and stir gently to combine.
Serve. Ladle into bowls and top with reserved crab, a tiny pinch of flaky sea salt, microgreens (or fresh herbs), and freshly ground black pepper. For an elegant finish, drizzle about 1 tsp heavy cream over each bowl. Serve immediately while hot.