1/2 pound skinless, boneless salt cod (baccalà), rinsed well
5 tablespoons extra-virgin olive oil plus more for drizzling
1 large shallot, minced
1 1/2 pounds butternut squash, peeled, seeded and cut into 1/4-inch pieces (4 cups)
1 medium Yukon gold potato, peeled and cut into 1/4-inch pieces
Coarse sea salt
1/2 cup dry white wine
1 tablespoon finely chopped parsley
1 small garlic clove, finely chopped
12 large shrimp, peeled and deveined
Freshly ground black pepper

Cover cod with 2 inches of cold water in a large bowl and soak, chilled, changing water 3 times a day, until a small piece tastes pleasantly salty but not overwhelming, 1 to 3 days.

In a large saucepan, heat 2 tablespoons oil over medium heat; add shallot and cook, stirring frequently, for 2 minutes. Add squash and potato; cook, stirring occasionally, until vegetables begin to soften, about 10 minutes. Add water to just cover vegetables (about 2 cups) and pinch salt. Bring to a gentle simmer and cook until vegetables are very tender, about 15 minutes. In batches, carefully purée soup in a blender until smooth. Return soup to pot.

Drain cod and combine it with wine, remaining 2 tablespoons oil, parsley and garlic in a medium saucepan. Bring to a bare simmer over medium-low heat. Reduce heat to very low and cook, covered, for 10 minutes. Strain cooking liquid through a fine-mesh sieve into a bowl; stir 1/2 cup into soup. Thin soup with water or more cooking liquid to taste, to desired consistency.
Cut cod into bite-size pieces.

Bring soup to a simmer. Add shrimp and cook until shrimp are opaque, about 3 minutes. Remove pot from heat. Gently stir in cod. Season soup to taste with salt and pepper. Serve drizzled with oil.