Combine lamb, salt, pepper, and oregano in a medium bowl. Mix with hands until homogeneous. Cover and refrigerate at least 1 hour or up to overnight.

Meanwhile, place yogurt in a fine-mesh strainer set over a small bowl. Allow to drain in refrigerator for 1 hour.

Adjust oven rack to middle position and preheat oven to 300°F (150°C).

Place cold lamb mixture in bowl of food processor with onion, garlic, and bacon. Process until smooth a purée is formed, about 1 minute total, scraping down sides with rubber spatula as necessary.

Line rimmed baking sheet with aluminum foil. With moist hands, form meat mixture into rectangles about 1 ½-inches high, 8-inches long, and 5-inches wide.

Bake until center of loaf reads 155°F (68°C) on an instant-read thermometer, about 30 minutes. Allow loaf to rest at room temperature for 15 minutes (or refrigerate for up to a week).

Meanwhile, make the sauce. Combine strained yogurt, mayonnaise, garlic, lemon juice, and parsley and stir to combine. Season to taste with salt. Refrigerate, covered, until ready to use.

Adjust broiler rack to highest position (about 1 ½ to 2 inches from broiler element) and preheat broiler.

Slice loaf crosswise into ⅛ to ¼-inch strips (each strip should be about 5-inches long and 1 ½-inches wide). Lay strips on rimmed baking sheet lined with aluminum foil and broil until edges are brown and crispy, about 2 minutes.

Tent with aluminum foil. Warm bread by placing on rimmed baking sheet and broiling until soft and pliant, about 45 seconds per side.

Spread ¼ cup of sauce over each piece of bread. Divide meat evenly between sandwiches. Top as desired, wrap with foil, and serve.