1Toss potatoes with 1 Tbsp. of oil & some salt & pepper. Place in a bowl. Put beets in a separate bowl.
2Warm 1 Tbsp. of oil in a large nonstick skillet over medium heat. Add onion and sprinkle with salt; sauté 3 minutes. Add garlic; sauté for 1 minute. Stir into bowl with beets.
3Add 1 Tbsp. of oil to skillet. Saute potatoes until golden, 10 minutes. Stir in milk, scrape bits off bottom of skillet; boil until most of the milk has evaporated, 1 minute.
4Add beet mixture, thyme & 1/2 of parsley to skillet; sauté 5 minutes. Remove from heat, sprinkle with cheese. Season with salt & pepper; toss well.
5In a separate non stick skillet, fry eggs to desired doneness. I prefer mine...dippy. Divide hash among 4 bowls or plates (your choice), sprinkle with remaining parsley, top each with an egg and serve