How to Make It
Place 2 tablespoons water and potatoes in a microwave-safe dish; cover with plastic wrap. Microwave at HIGH 5 minutes or until tender. Place potatoes on a paper towel-lined plate.
Heat a large cast-iron skillet over high. Add 1 tablespoon oil to pan; swirl to coat. Add 1/2 teaspoon salt, red pepper, and chicken; cook 5 minutes, stirring to crumble. Remove chicken mixture to a bowl with a slotted spoon. Add onion and poblano chile to drippings in pan; cook 4 minutes. Add the mushrooms, 2 teaspoons thyme, and garlic; cook 5 minutes. Add mushroom mixture to chicken mixture.
Add remaining 1 tablespoon oil to pan; swirl to coat. Add potatoes; cook 4 minutes, stirring occasionally. Add remaining 1/2 teaspoon salt, chicken mixture, and 1 tablespoon vinegar to pan; cook 2 minutes.
Add water to a saucepan, filling two-thirds full; bring to a boil. Reduce heat; add remaining 2 tablespoons vinegar. Break each egg into a custard cup, and pour each gently into pan; cook 3 minutes. Carefully remove eggs from pan using a slotted spoon.
Divide hash evenly among 4 plates; top with eggs and remaining 1 teaspoon thyme.