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Preparation Bring a medium pot of water to a boil. Carefully slip in eggs and cook 8 minutes. Transfer to a bowl of ice water with a slotted spoon; let cool. Peel eggs, then slice each crosswise into 5 pieces.
Meanwhile, mix yogurt, lemon juice, and garlic in a small bowl; season yogurt sauce with salt.
Heat oil in a large skillet over medium-high. Add sausage and press down on it with a wooden spoon or heatproof rubber spatula to flatten; cook, undisturbed, until browned underneath, about 4 minutes. Turn sausage over; break into smaller pieces with spoon. Cook until cooked through and crisp, about 3 minutes. Transfer sausage along with drippings to a medium bowl. Reserve skillet.
Lay out lavash on a work surface. Spread 1/4 cup yogurt sauce over each, leaving a 1" border. Top with sausage, dividing evenly. Top each with 10 egg slices in a single layer, then with red onion, pickles, and mint. Fold in 2 sides of lavash and, starting at an unfolded edge, roll up tightly.
Set reserved skillet over medium heat and cook 2 wraps, undisturbed, until golden brown and starting to crisp underneath, about 2 minutes. Turn over and cook until golden brown and starting to crisp on the other side. Transfer to a cutting board. Repeat with remaining wraps. Cut wraps in half crosswise.