Bring a large pot of water to a boil. (For 6 eggs, use about 2 qt. water.) Using a slotted spoon or tongs, gently lower eggs, one at a time, into water. Return to a boil, cover, and reduce heat to medium-low. Cook for 12 minutes.
A few minutes before eggs are cooked, fill a large bowl with about 2 cups ice and about 3 cups water. Transfer eggs to ice bath. Let sit for about 30 seconds.
Using the back of a spoon, tap each egg, making sure to tap all over shell. (Alternatively, you can tap and roll the eggs on a counter; keep in mind not to tap the eggs too hard!) Return cracked eggs to ice bath and let sit 5 minutes. Peel eggs under a thin stream of cool running water, or peel while still in ice water.
Coarsely chop eggs, then transfer to a large bowl. Add mayonnaise, dill, mustard, relish, hot sauce, sour cream, and paprika; season with salt and pepper. Taste and add more hot sauce, if needed.
Transfer dip to a serving bowl. Sprinkle with more dill and paprika. Serve with crackers alongside.
