Add the crawfish to a large pot of salted boiling water. Cook for about 30 seconds. Shock the crawfish in a bowl filled with ice water.
Cut through the shell on the belly side the entire length of the crawfish. This will help the meat cook evenly. Remove the vein along the body as well as the sack behind the eyes. Rinse thoroughly.
Combine the butter, hot sauce, tarragon and cayenne in a food processor. Process until smooth. Season with salt and pepper.
Prepare a charcoal grill for direct grilling. Place the crawfish in a grill basket and grill until the shell turns bright red and the meat is opaque, about 5 minutes, shaking the grill basket occasionally.
Meanwhile, melt the spicy tarragon butter in a saucepan or skillet. Remove the crawfish to a large platter or shallow bowl, season with salt and pepper and drizzle with the melted butter. Serve immediately.
Select designates 12 to 15 crawfish per pound; True Select designates 12 crawfish or less per pound.