Place the eggs in a saucepan and cover them with water. Bring to a boil over medium-high heat.

Once boiling, cover the pan, remove it from heat, and let it sit for 12 minutes.

While the eggs are cooking, combine the sliced beet, water, vinegar, sugar, and salt in a small saucepan. Bring to a boil, then reduce heat and simmer for 10 minutes. Remove from heat and let cool.

Once the eggs are done, transfer them to an ice bath for 5 minutes to cool. Peel the eggs and carefully slice them in half lengthwise. Remove the yolks and place them in a mixing bowl.

Add the mayonnaise, Dijon mustard, black pepper, and half of the chopped smoked salmon to the yolks. Mash together until smooth and creamy.

In a separate bowl, combine the beet mixture and the egg whites. Let the egg whites soak in the beet juice for at least 30 minutes to achieve a vibrant color.

Fill the beet-pickled egg whites with the yolk mixture. Top with the remaining smoked salmon, chopped dill, and a few capers for garnish.

Serve immediately or refrigerate until ready to serve.