Heat olive oil in a saucepan over medium-high heat. Add onion, salt, and lamb. Break up lamb into small crumbles. Cook, occasionally stirring, until most of the liquid evaporates, about 8 minutes. Toss in garlic, currants, pine nuts, cumin, coriander, cinnamon, paprika, black pepper, cayenne, and allspice. Cook and stir for 1 minute.

Pour tomato sauce into lamb mixture. Add water, stir, and reduce heat to medium. Cook until lamb mixture dries up and you can stir it without seeing liquid on the bottom of the pan, 20 to 30 minutes. Turn off the heat and let cool completely before using.

Combine egg, yogurt, water, and butter in a bowl. Whisk together thoroughly to make the egg wash.

Preheat the oven to 400°F (200°C). Butter a round baking pan or sheet pan.

Place 2 sheets of phyllo on your work surface. Keep the remaining sheets covered with a damp towel. Lightly brush some egg wash on top. Layer on 2 more sheets, one at a time, brushing more egg wash over each.

Line 1/3 of the lamb filling along one wide edge of the phyllo. Roll pastry up starting from the filling side and place it against the edge of the pan. Brush more egg wash on top.

Shape and fill 2 more rolls with the remaining phyllo, most of the egg wash, and filling. Wrap the rolls along and inside the first one, filling the pan, and brush with egg wash. Sprinkle sesame seeds on top.

Bake in the preheated oven until browned and crisp, 35 to 40 minutes. Let cool for 15 minutes.

Combine yogurt, mint, lemon juice, and garlic for the yogurt sauce. Mix in enough water to achieve the desired dipping consistency. Season with salt and cayenne.