Place 6 eggs in a saucepan, cover with cold water, bring to a full boil then reduce to a gentle boil and cook for 8–10 minutes.
Immediately transfer cooked eggs to a bowl of cold water or ice bath and cool completely for 10–15 minutes.
Peel all eggs carefully keeping egg whites as smooth and intact as possible and place in a bowl or container in a single layer.
Peel fresh beet and cut into small half-inch cubes then place in a medium saucepan with 6 cups water, 1 teaspoon salt, and juice and halves of one lemon.
Bring beet brine to a boil then reduce to a simmer and cook 10–15 minutes until water is deeply colored and beet cubes are tender.
Remove from heat and cool brine completely to room temperature — never pour hot or warm brine over eggs.
Remove beet cubes from cooled brine and set aside for another use if desired.
Pour cooled beet brine over peeled eggs ensuring they are completely submerged — add a small amount of extra water if needed to fully cover.
Cover container and refrigerate for a minimum of 3 hours for bright vivid pink or overnight for deeper more saturated magenta — do not soak longer than overnight.
Remove eggs from brine and pat gently dry with paper towels.
Halve each egg lengthwise with a clean sharp knife wiping blade between each cut for clean precise edges.
Carefully pop each yolk into a mixing bowl keeping pink egg white halves intact.
Add heaping tablespoon mayonnaise, Dijon mustard, pinch of salt, and pinch of black pepper to yolks and mash and mix until completely smooth and creamy.
If filling is too thick or crumbly add additional mayonnaise one small teaspoon at a time until smooth and pipeable.
Fill each pink egg white half using a small spoon for rustic presentation or a piping bag with a star tip for an elegant swirl.
Garnish immediately with fresh herbs, edible flowers, ground black pepper, or a dusting of ground chili or smoked paprika.
Arrange on a serving platter and serve chilled.