Place bulgur in a large bowl and cover with boiling water. Let stand until water is absorbed, about 30 minutes.
Preheat the oven to 350 degrees F (175 degrees C).
Stir 1/4 cup onion, 1 1/2 teaspoons salt, basil, and pepper into the bulgur. Add ground beef and 1/2 pound lamb; mix well. Divide mixture into 2 portions. Pat 1 portion over the bottom of a 12x7-inch baking dish.
Pat the other portion into a matching rectangle on a piece of waxed paper. Set aside for the top.
Cook and stir remaining 1/4 cup onion and 1/2 pound lamb in a skillet until browned, about 5 minutes. Drain excess grease from the skillet. Stir in pine nuts and remaining 1/2 teaspoon salt. Spoon lamb-pine nut mixture over the meat layer in the baking dish.
Invert top layer over the filling. Peel off waxed paper. Score top layer into diamond shapes by cutting diagonally in both directions, leaving about 3 inches between cuts. Press a small dab of butter into the middle of each diamond.
Bake in the preheated oven until golden brown, about 40 minutes. Garnish with cherry tomatoes and mint.
