Beat butter in a bowl until creamy; add sugar and continue beating until light and fluffy. Beat in egg yolk.
Whisk flour, allspice, and turmeric together in a bowl. Sift flour mixture into butter mixture and stir until dough is combined. Press dough into a 9-inch square baking dish. Beat egg white in a small bowl and brush dough with egg white.
Bake in the preheated oven until cake feels firm when lightly pressed in the center, 30 to 45 minutes. Cut into squares while cake is still hot; cool in baking dish.