Whisk mayonnaise, horseradish, vinegar, tarragon and tablespoon of dill in a small bowl. Season with salt, pepper, and more horseradish, if desired after tasting.
In a large, wide pot combine dill, shallot, peppercorns, bay leaf, and 3 tablespoons of salt in a large wide pot. Squeeze juice from lemon halves into the pot and add lemon halves as well. Add 3 quarts water and bring to a boil. Reduce heat to medium and let simmer for 10 minutes
Working in batches, drop a few shrimp in at a time until shrimp just begin to curl and turn opaque, about 30 seconds. Transfer to paper towel and let cool.
Chop up shrimp to desired size and toss with cup horseradish mayonnaise in a medium bowl until evenly coated. Season to taste with salt, pepper, and more horseradish, if desired. Can make this 8 hours ahead. Pile salad on toasted potato buns or on top of a green salad!