Place the egg in a small saucepan and cover with cold water by 1 inch. Bring to a rolling boil over medium-high heat, then cover, remove from heat, and let stand for 10 minutes. Transfer the egg to an ice bath until cool, about 5 minutes, then peel and finely chop.
Very finely mince the white onion.
In a medium bowl, whisk together the mayonnaise, ketchup, lemon juice, Worcestershire sauce, sweet paprika, kosher salt, and black pepper until smooth.
Fold in the sweet pickle relish, minced onion, and chopped egg until evenly distributed.
Cover and chill until the flavors meld and the dressing slightly thickens, 30–60 minutes. Stir, taste, and adjust salt or lemon juice to balance. Serve chilled.