Place the ground lamb in a large bowl. Add the grated onion, garlic, breadcrumbs, egg, parsley, red pepper flakes, cumin, salt and black pepper. Knead for 2 to 3 minutes until evenly mixed and slightly tacky.
Cover the lamb mixture and refrigerate for 20 minutes. This firms the filling and makes it easier to shape.
Preheat the oven to 190°C / 375°F. Divide the lamb mixture into equal portions and shape each into a long kebab log about 2.5 to 3 cm thick.
Lay a sheet of yufka or thin lavash on the work surface. Brush lightly with melted butter. Place a lamb log near the edge and roll tightly but gently. Repeat with the remaining lamb and bread.
Place the rolls seam-side down. Slice them into 3 to 4 cm pieces using a sharp knife. Arrange cut-side up on a parchment-lined baking tray and brush lightly with the remaining melted butter.
Bake for 25 to 30 minutes, until the bread edges are golden and the lamb filling reaches 71°C / 160°F internally.
Meanwhile, melt the butter with olive oil in a saucepan. Add tomato paste and cook for 1 minute. Whisk in hot water, salt and optional pepper flakes. Simmer for 3 to 4 minutes until smooth and glossy.
Mix the yogurt with grated garlic, if using. Spoon the yogurt into the centre of a warm serving plate.
Arrange the baked Beyti slices around the yogurt. Spoon warm tomato-butter sauce over the slices, sprinkle with parsley and serve immediately.