Place the beetroots in a pot with water to cover, and cook the beetroots for about 45 minutes, until tender.

In another pot, add vinegar, sugar, water, and spices, and bring to a boil.

Remove the pot with the beetroots from the heat, and let the beetroots sit in their cooking water to draw flavor and cool a bit.

Pour cold water over the beetroots and rub off the skins.

Cut the beetroots into 1 cm thick slices and add them to the vinegar brine.

Place the eggs in a pot, cover with water, and bring to a boil.

Once the water boils, let the eggs cook for 5 minutes, then pour cold water over them to stop the cooking. Peel the eggs, place them in the brine, and let them steep for at least 24 hours so the brine can flavor the eggs.

The eggs can stay in the brine for up to 3 weeks, and the brine can be reheated and reused.