For the cake, combine the chopped apple, 1 cup sugar, and 2/3 cup melted butter in a large bowl; stir well. Add eggs and vanilla, stirring well.
Combine flour and the next 4 ingredients; add to the apple mixture, stirring well. Stir in the pecans. Spoon the batter into a greased 13x9x2 inch baking pan. Bake at 350 degrees for 45 minutes or until a wooden pick inserted in the center comes out clean. Let cool in the pan on a wire rack.
For the frosting, combine the cream cheese and 1/4 cup softened butter, beating well at medium speed of an electric mixer. Add the lemon juice and powdered sugar; beat until light and fluffy. Spread frosting on top of the cooled cake.