For the tarragon mayonnaise: Whisk together the egg yolk, lemon juice, mustard and 1/2 teaspoon salt in a medium bowl. Very, very slowly add the oil while whisking constantly (the first few additions should only be about 5 drops of oil). Continue to add the oil, whisking constantly, until all of the oil is incorporated and emulsified. Add the tarragon and cover. Let stand at room temperature for 2 hours, then refrigerate until cold.
For the BLTs: Meanwhile, bring a medium pot of well-salted water to a boil over medium-high heat. Squeeze the juice from half of the lemon and add it to the water. Add the shrimp and cook until just cooked through, about 2 minutes. Drain; cool slightly, then refrigerate until just no longer warm. Coarsely chop the shrimp (remove the tails) and transfer to a medium bowl. Add the celery and about two-thirds of the tarragon mayonnaise (or to taste). Season with salt, pepper and additional lemon juice to taste. (Any remaining mayonnaise can be kept refrigerated in an airtight container up to 1 day.)
Place the bacon and tomatoes in the buns; sprinkle the tomatoes lightly with salt, then fill the buns generously with the shrimp salad. Top with the arugula and a light squeeze of lemon. Serve immediately.