Combine yogurt and mango chutney in a bowl; stir well, and set chutney sauce aside.
Cook meat and onion in a large nonstick skillet over medium heat until browned, stirring to crumble. Drain meat mixture in a colander; wipe drippings from skillet with a paper towel.
Return meat mixture to skillet. Add currants and next 6 ingredients (currants through black pepper); stir well. Combine water and cornstarch; stir well, and add to the meat mixture. Cook over medium heat 2 minutes, stirring constantly. Remove from heat; set aside.
Working with 1 phyllo sheet at a time, cut each sheet lengthwise into 4 (3 1/2-inch-wide) strips; lightly coat strips with cooking spray (cover remaining phyllo dough to keep it from drying out). Stack 2 strips, and spoon about 1 tablespoon meat mixture onto one end of each stack. Fold the left bottom corner over mixture, forming a triangle; keep folding back and forth into a triangle to end of strip.
Place triangles, seam sides down, on a baking sheet; lightly coat with cooking spray. Bake at 400° for 12 minutes or until golden. Serve warm with chutney sauce.