1. Preheat oven to 400 degrees. Steam potatoes and beets in a steamer basket set over a pot of simmering water, covered, until easily pierced with the tip of a knife, 12 to 15 minutes; drain.

2. Heat a large ovenproof skillet over medium-high. Swirl in oil. Cook onion, stirring occasionally, until translucent, 3 to 4 minutes. Stir in butter until melted. Stir in garlic, potatoes, and beets until evenly coated; season with salt and pepper. Mash some potatoes against side of pan with the back of a spoon. Stir in pastrami and press mixture into bottom of skillet.

3. Transfer hash to oven; roast until top is crisp in places, 20 to 25 minutes. Make 6 wells in hash with the back of a spoon. Crack 1 egg into each well. Return to oven; cook until whites are just set but yolks are still runny, 6 to 8 minutes. Meanwhile, toss carrots, parsley, dill, and vinegar in a bowl; season with salt. Serve with salad, toast, and butter.