For an intense pickled flavor, let the eggs cool slightly, then peel them before submerging themin the beet or turmeric mixture.
Bring beets and enough water to cover them by 2 inches (about 10 cups) to a boil in a medium saucepan. Reduce to a simmer; cook until beets are tender when pierced with the tip of a knife, about 45 minutes. Drain, discarding liquid and reserving beets. When cool enough to handle, rub skins off with a paper towel and discard. Coarsely chop beets and place in a large bowl; stir in 3 cups water and vinegar.
Meanwhile, bring turmeric and 6 cups water to a boil in a medium saucepan. Reduce to a simmer; cook until water is a bright, deep yellow, about 30 minutes. Strain into a large bowl; discard solids. Stir in vinegar; let cool completely.
Bring a large pot filled halfway with water to a boil. Using a slotted spoon, gently lower eggs into water. Reduce heat to a gentle simmer and cook 9 minutes. Use slotted spoon to remove eggs; transfer half to bowl of turmeric dye and other half to bowl of beet dye. Refrigerate until desired color is achieved, at least 15 minutes, and up to 3 hours for more saturated color. Remove eggs from dye with slotted spoon; refrigerate until ready to use.
Peel and halve eggs length-wise; remove yolks and transfer to a bowl. Mash yolks with a fork; stir in mayonnaise and season with kosher salt and pepper. Spoon yolk mixture into whites. Combine grated beet and horseradish with vinegar and a pinch of kosher salt. Garnish filled eggs with beet mixture or ramp pesto, or both. Sprinkle with flaky salt and pepper; serve.