Season the shrimp with salt and white pepper and toss to coat.
Add butter, shallot, and tarragon to a large skillet over low heat. Once melted and fragrant, add the shrimp in an even layer.
Poach gently for 3–4 minutes, stirring occasionally, until the shrimp are just opaque and cooked through. Do not let the butter boil.
Remove from heat and let the shrimp sit in the warm butter for 10 minutes before transferring to a cutting board. Discard the shallot and tarragon.
Chop the shrimp into a mix of larger and smaller pieces, lightly smashing some for texture. Chill completely.
In a bowl, combine chilled shrimp with mayonnaise, chives, lemon juice, and tarragon. Season with salt and white pepper to taste.
Butter the outsides of the buns and griddle in a hot pan until golden and crisp on both sides.
Fill each bun with the chilled shrimp salad and serve immediately.