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.
I have discovered a new food to love: sunchokes! They are sweet, nutty and earthy in a fantastically unique way. However, they are not kidding about buying the least knobby sunchokes possible. Peeling the knobby ones I bought was an absolute pain and took forever. It's such an awful job I'd go so far as to say don't even bother to buy them unless the sunchokes are relatively smooth!
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