Heat a skillet on medium heat. Heat oil in the pan. Add paprika and let it sizzle for 15 seconds to bring out the flavor, then add garlic, shallot, and thyme. Sauté for 2-3 minutes until fragrant.
Add sweet potato, Yukon gold potato, beets, and carrots. Season with salt and pepper. Cook for 2-3 minutes. Then, add water and cover. Reduce heat to medium-low and steam for 10 minutes.
Remove cover and continue to cook until caramelized, 3-5 minutes
Make four wells in the hash and crack an egg into each. Re-cover and cook for 6-8 minutes, or until egg whites are set and yolks are cooked to your liking.
Garnish with fresh parsley; serve warm.