Heat the oven to 425°F (220°C). Put potatoes in a saucepan, cover with water by about 1 inch and bring to a boil. Simmer 18–25 minutes, until a knife passes easily through the centers. Drain thoroughly and let steam-dry 5 minutes.

Brush two rimmed baking sheets with 1 tablespoon of the oil. Space the potatoes apart. Press each with a flat-bottomed glass to about 1/4 inch thick, keeping the pieces together where possible. Brush with the remaining oil and sprinkle with 1/2 teaspoon salt and the pepper.

Roast 30–40 minutes, switching the trays between racks halfway through, until the undersides and ragged edges are deep golden and crisp. Let stand about 10 minutes, until warm rather than piping hot.

Meanwhile, stir the yogurt, mustard, lemon juice, zest, garlic and remaining 1/4 teaspoon salt in a bowl. Pat the cucumber dry and add it with the scallions and dill. Taste and adjust lemon or salt if needed.

Break large potatoes into bite-sized pieces. Fold most into the dressing just before serving. Scatter the remaining crisp pieces over the top and serve immediately.

Refrigerate leftovers promptly in a shallow covered container at 40°F (4°C) or below; within 2 hours, or 1 hour above 90°F. Use within 3–4 days. Dressed potatoes soften in storage.