Step 1

Place beet juice, vinegar, water, sugar, peppercorns, and 2 teaspoons of the salt in a small saucepan. Bring to a boil over high, stirring occasionally to dissolve sugar. Remove from heat, and cool 25 minutes.

Step 2

Place eggs in a 1-quart canning jar or medium-size container. Pour vinegar mixture over eggs. Loosely cover with lid; chill at least 12 hours or up to 24 hours. (If needed, press plastic wrap directly onto surface of eggs to keep submerged.)

Step 3

Drain eggs; pat dry with paper towels. Cut eggs in half lengthwise. Carefully remove yolks; place in a medium bowl. Set aside egg white halves. Mash yolks using a fork; add mayonnaise, cornichons, horseradish, dill, mustard, and remaining ¼ teaspoon salt, and stir until smooth. Transfer yolk mixture to a ziplock plastic bag, and seal. Snip a hole in tip of plastic bag. Pipe about 1 tablespoon yolk mixture onto each egg white half. Garnish with dill.