To boil eggs: Place eggs in a pot and cover the eggs with water. Transfer the pot to the stove and turn the heat to high. Bring the pot of water to a medium boil. The second the boil begins, turn the heat off, cover the pot and set your timer to 9 minutes. This will give you a firm yolk---yet not overcooked---that's perfect for deviled eggs. Drain the eggs and run cold water on them--just so they're cool enough to handle. I find that warm eggs are easier to peel than cold eggs. Peel each hard boiled egg and set them aside.
To a medium saucepan, add all of the pickling ingredients: water, apple cider vinegar, sugar, salt, beet juice, beet, yellow onion, garlic clove, cloves and cinnamon stick. Bring the mixture to a simmer; cook for 20 minutes, until the beet is very tender when poked with a fork.
Transfer the pickling liquid to the refrigerator or freezer (I put it in the freezer for 15 minutes to cool it down quickly) to come to room temperature. (We're doing this so the liquid doesn't overcook our perfectly boiled eggs.) When the mixture is room temperature, transfer it to a large bowl or jar. Carefully drop in the hard boiled eggs and allow to pickle for at least 2 hours or up to 2 days. Obviously, the longer you allow them to sit, the more pink in color they'll be and the more they'll taste like the pickling liquid. The pickled eggs you see pictured were pickled for 2 1/2 hours.
Once pickled, remove the eggs from the pickling liquid and slice them in half. Gently scoop out the egg yolks and transfer them to a medium bowl. Mash the egg yolks with a fork until they reach a fine crumble. Mix in the mayonnaise, sour cream, Dijon mustard and freshly ground pepper until the egg yolks are very smooth. Salt to taste and adjust any other seasonings to your liking (you may find you need to add a tablespoon or two more of mayo or sour cream to reach a smooth consistency).
Scoop dollops of the mixture into each of the egg-white halves. Alternatively, if you're feeling fancy, transfer the mixture to a piping bag fitted with a star tip and pipe-in the mixture into of the each egg white halves. Garnish with a teeny sprig of fresh dill, if you like, and serve.