Place whole beets in a pot and cover with water. Bring water to a boil, cover the pot, reduce heat to medium-low, and cook until beets are tender, about 30 minutes.
Place carrots in a microwave-safe dish and cook on high until firm but tender, about 6 minutes. Place carrots in a colander under cold water to stop the cooking process, 3 to 5 minutes. Remove peels and discard; cut beets into thin slices.
Whisk cider vinegar, sugar, ginger, orange zest, dill, sea salt, garlic powder, mustard, and black pepper together in a microwave-safe bowl until sugar and salt dissolve. Whisk oil into cider dressing mixture. Microwave dressing on high for 2 minutes; whisk again.
Combine beets, carrots, and onions together in a large container; pour dressing on top and mix well. Cover the container and refrigerate for at least 24 hours before serving.