Drain the canned beets into a bowl, reserving 1 cup of the beet liquid; set the beets aside.
Place 6 peeled hard-boiled eggs into a clean 2-quart glass jar along with some of the reserved beets, 6 whole garlic cloves, 1/8 teaspoon whole peppercorns, and a pinch of salt.
In a small saucepan combine 3/4 cup apple cider vinegar, 1 1/4 cups water, 1 cup white sugar, and the reserved 1 cup beet liquid.
Bring the mixture to a boil, then reduce heat and simmer 1–2 minutes until the sugar dissolves.
Carefully pour the hot pickling liquid over the eggs and beets in the jar so the ingredients are fully submerged, then let the jar cool to room temperature.
Screw the lid on tightly and refrigerate for at least 24–48 hours before serving to allow flavor and color to develop.
For the lemon-rosemary variation, place eggs in a jar with peels from 1 lemon, 5 sprigs rosemary, 6 cloves garlic, 1/8 teaspoon peppercorns, and a pinch of salt; prepare the same pickling liquid without beet juice and pour hot over the eggs.
For the vegetable-dill variation, place eggs with 4 cloves garlic, 6 carrot rounds, 6 zucchini rounds, 5 sprigs fresh dill, 1/8 teaspoon whole pepper and a pinch of salt; pour the same hot vinegar-sugar-water brine over the eggs and vegetables and cool, then refrigerate 24–48 hours.
