For the quick pickle: Cut the stems and leaves off the top of each beet; save the root for the hash. Strip the leaves from the beet stems and tear the leaves into bite-size pieces. Set the leaves aside. Cut the stems into thin half-moons and place in a jar or small bowl.
To the jar or bowl with the beet stems add the vinegar, honey, salt, and jalapeño. Stir or shake well to dissolve the honey and salt. Allow the pickle to sit for at least 30 minutes or up to 1 week in the refrigerator.
For the hash: On a rimmed baking sheet place the beets, sweet potatoes, shallots, onions, and garlic. Broil the vegetables, flipping often, until just tender when pierced with the tip of a knife, about 25 to 30 minutes. Some vegetables may cook faster than others so remove them as they cook. Allow the vegetables to cool enough to handle. Peel the beets, sweet potatoes, onion, and garlic cloves. Discard the skins. Cut the vegetables into large pieces, each about 1 to 2 inches.
Heat a 10-inch cast-iron skillet over medium-high heat. Add 2 tablespoons of the butter to the skillet and heat until melted. To the hot pan add the beets, sweet potatoes, shallots, onions, and garlic cloves. Stir to combine. Spread the vegetables evenly in the pan and cook on one side until they begin to brown in spots, about 5 minutes. Flip the vegetables and continue to cook until the other side begins to brown, another 5 minutes. Season with 1/2 teaspoon salt and stir to combine. Add in the torn beet leaves and stir gently to wilt, about 1 minute.
Clear a few divots in the hash and add the remaining tablespoon of butter. Crack the desired number of eggs into the divots. Season the eggs with the remaining 1/2 teaspoon of salt and the pepper. Cook the eggs until the whites are set but the yolks are still runny, about 4 minutes, or cook to the desired doneness. Serve warm garnished with the pickled beet stems and chives.