Preheat oven to 325 degrees. Grease a 9x13-inch baking dish.

Beat white sugar and shortening together in a bowl using an electric mixer until creamy; add eggs and mix well. Mix cinnamon, cloves, and vanilla extract into creamed butter mixture.

Whisk buttermilk and baking soda together in a separate bowl; stir into creamed butter mixture until thoroughly mixed. Slowly stir flour into butter-buttermilk mixture until batter is smooth; fold in blackberries. Pour batter into the prepared baking dish.

Bake in the preheated oven until a toothpick inserted in the center of the cake comes out clean, about 40 minutes. Cool in the pan for 10 minutes before removing to a serving plate.

Combine brown sugar, evaporated milk, and butter in a saucepan; cook and stir mixture over medium heat until a small amount of brown sugar mixture dropped into cold water forms a soft ball. Remove saucepan from heat and stir in vanilla extract. Beat by hand until icing is thick. Spread icing quickly over top and sides of cake.