Add olive oil to a large cast iron skillet or sauté pan. Add chopped beet and sweet potato. Sauté for about 15 minutes, stirring every so often, until sweet potato and beet give slightly when pricked with a fork. They will continue to cook, so be sure not to cook them too much.
Meanwhile, fill a large pot with water for the poached eggs. Allow to come to a simmer while you prep the other ingredients.
When sweet potato and beets are just about tender, add the onion, pepper, and garlic. Sauté, stirring often, until all vegetables are cooked through. This should be another 10 minutes or so.
When water for eggs is simmering, add a good glug (about 1 tablespoon) of distilled white vinegar. I find that this helps the whites of the eggs set. Crack 1 egg in to a small dish. Use a slotted spoon to stir the water, creating a vortex. Gently pour the egg in to the water, allowing the egg to swirl into itself. Continue with other eggs, but be sure not t overcrowd the pot.
Poach eggs for 3-4 minutes. Using a slotted spoon, remove egg and press gently with your finger. The egg should slightly give; you want the yolk "jammy", not runny or over cooked.
To serve, dish hash on to 4 plates, topping each with a poached egg. Garnish with rosemary for a festive touch, and serve with a variety of hot sauces.