How to Make It
Combine 8 cups water and lemon juice in a large pot; add sunchokes and potato. Bring to a boil over high heat. Simmer 35 minutes or until sunchokes and potato are tender when pierced with a fork; drain. Cool. Peel sunchokes with a sharp paring knife. Dice sunchokes and potato.
Preheat oven to 200°.
Heat a large cast-iron skillet over medium-high heat. Add 1 tablespoon oil to pan. Add sausage; cook 5 minutes, stirring to crumble. Add 1 1/2 teaspoons oil, onion, bell pepper, and garlic; cook 5 minutes. Transfer mixture to a plate.
Wipe pan clean with paper towels. Add remaining 1 1/2 tablespoons oil to pan. Add sunchokes and potato; cook over medium-high heat 7 minutes, stirring occasionally. Add sausage mixture, paprika, 1 teaspoon thyme, 1/2 teaspoon lemon rind, salt, and pepper. Keep warm in 200° oven.
Fill a saucepan with water to about 2 inches. Bring to a gentle simmer over medium heat; add vinegar to pan. Working with 1 egg at a time, crack each egg into a small ramekin; gently pour into pan. Cook eggs 3 1/2 minutes. Transfer poached eggs to a paper towel-lined plate using a slotted spoon. Divide hash among 6 plates; top each with a poached egg. Sprinkle evenly with remaining 2 teaspoons thyme and remaining 1 teaspoon lemon rind.