Bring a large pot of water to a boil. (For 12 eggs, use about 4 quarts water; work in batches if needed.) 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 4 cups ice and about 6 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. Scoop out yolks and transfer to a medium bowl.
To bowl with yolks, add mayonnaise, lemon juice, horseradish, tomato paste, hot sauce, and celery seed and mix until well combined; season with salt and pepper. Transfer to a piping bag fitted with a large tip (or snip a small corner off a plastic bag).
Place Old Bay seasoning on a small plate. Press cut side of eggs into seasoning.
Arrange eggs on a plate and pipe filling onto each egg. Top with a pickle slice and celery leaves.
