1. Prepare the Shrimp Poaching Liquid: In a large pot, combine the four cups of water, the halved lemon, bay leaf, and Old Bay seasoning. Bring this liquid to a rolling boil over high heat.

2. Poach the Shrimp: Once the water is boiling rapidly, add the raw shrimp. Cook for two to four minutes until they are opaque and pink. Immediately remove them from the heat and drain them thoroughly.

3. Chill the Shrimp: Transfer the cooked shrimp to an ice bath to stop the cooking process instantly. Once fully chilled, drain them and pat them completely dry with paper towels. Refrigerate until ready to dress.

4. Assemble the Base: In a medium-sized bowl, whisk together the mayonnaise, mustard, ketchup, prepared horseradish, Worcestershire sauce, and lemon juice until the mixture is completely uniform and smooth.

5. Fold in Aromatics and Spices: Gently incorporate the smoked paprika, cayenne pepper, minced garlic, finely minced celery, sliced green onions, and chopped parsley. Stir only until just combined.

6. Season and Rest the Sauce: Season the remoulade generously with salt and black pepper. Taste and adjust seasoning as necessary. Cover the bowl and refrigerate the remoulade for a minimum of four hours.

7. The Final Dress: Just before serving, gently toss the chilled, dried shrimp with the prepared remoulade sauce until every piece of seafood is thoroughly coated.

8. Present the Dish: Arrange the dressed shrimp on a chilled platter or individual serving plates, often garnished with fresh lemon wedges and extra parsley.