Set a steamer basket over boiling water. Gently set eggs in basket, cover, and set a timer for 9 minutes. Meanwhile prepare a large bowl of ice water (the colder the water, the easier the eggs will peel, so be generous with the ice). When timer dings, run eggs briefly under cold water and then gently set in ice water. Chill eggs at least 30 minutes, then keep in fridge until ready to peel.
Set out 2 clean quart jars. In packing the jars, you'll want to keep the eggs from smooshing against the side, creating a white spot. Carefully layer red onions, cooked beets, eggs, and 2 sprigs dill in each jar, making sure eggs are surrounded by other vegetables. Carefully pour in brine to cover. Seal jars and refrigerate overnight, or up to a week. Note that the color will nearly saturate the eggs after 2 days.
Serve eggs sliced in half (you'll want to clean the knife between slices) alongside pickled beets and red onions, dollop with horseradish sauce, and sprinkle with minced herbs and fresh ground pepper.