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1. Peel and coarsely shred potatoes. Place potatoes in a large bowl; add enough water to cover potatoes. Stir well. Drain in a colander set over the sink. Repeat rinsing and draining two or three times until water runs clear. Drain again, pressing out as much water as you can with a rubber spatula. Line a salad spinner with paper towels; add potatoes and spin.* Repeat, if necessary, until potatoes are dry. In a large bowl combine potatoes, corned beef, carrot, beets, thyme, 1/4 teaspoon salt, and 1/4 teaspoon pepper.
2. In a 10-inch nonstick skillet** heat 1 tablespoon of the oil and the butter over medium-high heat until butter foams. Add potato mixture to the skillet, spreading into an even layer. Gently press with the back of a spatula to form a cake. Reduce heat to medium-low. Cook, without stirring, about 14 minutes or until the bottom is golden brown and crisp.
3. Invert a plate over the top of the skillet. Carefully invert the skillet to transfer the hash mixture to the plate. If needed, add the remaining 1 tablespoon oil to the skillet. Using the plate, slide the hash mixture back into the skillet, uncooked side down. Cook about 8 minutes more or until the bottom is golden brown. Sprinkle with blue cheese. Transfer to a serving platter. Cover loosely and keep warm.
4. Wipe out the skillet and fill it halfway with water. Add vinegar and bring to boiling; reduce heat to simmering (bubbles should begin to break the surface of the water). Break an egg into a cup and slip egg into the simmering water. Repeat with the remaining eggs, allowing each egg an equal amount of space in the water-vinegar mixture. Simmer eggs, uncovered, for 3 to 5 minutes or until whites are completely set and yolks begin to thicken but are not hard. Using a slotted spoon, remove eggs from water. (Season to taste with additional salt and pepper.)
5. Top each serving of hash with a poached egg.