Toss potatoes with 1 Tbsp. oil and 1/4 tsp. each salt and pepper. Place beets in a separate bowl.
Warm 1 Tbsp. oil in a large nonstick skillet over medium heat. Add onion and sprinkle with salt; sauté 3 minutes. Add garlic; sauté 1 minute. Stir into bowl with beets. Add 1 Tbsp. oil to skillet. Sauté potatoes until golden, 10 minutes. Stir in milk, scrape bits off bottom of skillet; boil until almost evaporated, 1 minute.
Add beet mixture, thyme and half of parsley to skillet; sauté 5 minutes. Remove from heat, sprinkle with cheese. Season with salt and pepper; toss well.
In a separate nonstick skillet, fry eggs to desired doneness. Divide hash among 4 bowls, sprinkle with remaining parsley, top each with an egg and serve.