In a medium bowl, whisk the mayonnaise, ketchup, white vinegar, sugar, Worcestershire sauce, hot sauce, kosher salt, and black pepper until smooth.
Grate 2 tbsp onion directly into the bowl, then add the sweet pickle relish and the finely chopped hard-boiled egg; fold until evenly distributed.
Cover and chill for 30 minutes to let the flavors meld. Stir, taste, and adjust salt or vinegar to balance. Serve as a salad dressing or sandwich spread; refrigerate in an airtight container for up to 1 week.