Ingredients
Method
Heat ¼ cup extra-virgin olive oil in a medium Dutch oven or other heavy pot over medium. Cook reserved shrimp shells from 2 lb. large shrimp, peeled, deveined, cut into ½" pieces, shells reserved, stirring often, until pink and browned in spots, 6–8 minutes. Add 3 medium onions, finely chopped, 1 large russet potato (about 14 oz.), peeled, cut into 1" pieces, 3 celery stalks, chopped, 2 medium carrots, peeled, chopped, 10 garlic cloves, smashed, and 2 bay leaves and cook, stirring often, until vegetables are softened and onion is browned in spots, 8–10 minutes.
Add ⅓ cup double-concentrated tomato paste and cook, stirring often, until slightly darkened in color, about 5 minutes. Add 1 cup dry white wine and ½ cup cognac and bring to a boil. Cook, stirring often, until reduced by about two thirds, about 4 minutes. Add 8 cups low-sodium vegetable broth and bring to a boil. Reduce heat to medium; simmer 20 minutes.
Working in batches, transfer soup to a blender and blend on medium-high speed until very smooth. Strain through a fine-mesh sieve lined with a layer of cheesecloth back into pot (use a chinois if you have one and skip the cheesecloth); discard solids. Stir ½ cup heavy cream and 1½ tsp. Diamond Crystal or 1 tsp. Morton kosher salt into soup. Season with more salt if needed and freshly ground pepper. Set over low heat to keep warm.
Remove zest from 1 lemon in wide strips with a vegetable peeler, leaving as much white pith behind as possible and thinly slice zest strips. Halve lemon and squeeze juice into a small bowl.
Heat remaining ¼ cup extra-virgin olive oil in a medium skillet over medium-high. Cook sliced lemon zest and 6 garlic cloves, thinly sliced, stirring often, until softened and fragrant, 2–3 minutes. Add shrimp and cook, stirring often, until cooked through, 1–2 minutes. Add 2 Tbsp. unsalted butter and lemon juice; cook, stirring often, until sauce is emulsified and thickened slightly, 1–2 minutes. Remove from heat; stir in ¼ cup finely chopped parsley and season with salt and pepper.
Ladle soup into bowls and top with shrimp mixture. Do Ahead: Soup (without shrimp topping) can be made 1 day ahead. Cover and chill.