Place the eggs in a single layer in a saucepan and cover with cold water by one inch. Bring to a boil over medium-high heat, then remove from heat, cover, and let sit for 11 minutes.
Transfer the eggs to an ice bath and let cool completely, about 10 minutes. Peel carefully and slice each egg in half lengthwise.
Pop the yolks into a bowl and set the whites on a serving platter.
Mash the yolks well with a fork, then mix in the mayonnaise, Dijon, tarragon, and chives. Season with salt and white pepper to taste. The filling should be smooth, creamy, and well-seasoned.
For the crispy capers: heat oil in a small skillet over medium-high heat. Pat the capers completely dry, then fry for 1–2 minutes until they bloom open and turn golden and crisp. Transfer to a paper towel to drain.
Spoon or pipe the filling into the egg white halves.
Top each egg with a crispy caper or two, a pinch of fresh chives, and a small tarragon leaf. Serve immediately or refrigerate until ready to serve.