Cream butter in a large mixing bowl; gradually add brown sugar, beating until light and fluffy. Add eggs, one at a time, beating well after each addition. Add molasses; beat well.
Combine flour, soda, salt, and ginger; stir well. Add to creamed mixture alternately with boiling water, beginning and ending with flour mixture. (Batter will be thin.) Pour half of batter into a greased 13- x 9- x 2-inch baking dish; arrange half of apples over batter. Sprinkle with 1 tablespoon sugar, and dot with 1 tablespoon butter. Repeat procedure with remaining batter, apples, sugar, and butter.
Bake at 350° for 45 minutes or until a wooden pick inserted in center comes out clean. Cut into 3-inch squares. Transfer to serving plates; serve warm with ice cream.