Preheat oven to 350°F. Coat a 10-by-14-inch sheet-cake pan with nonstick spray.
To make cake: Toast walnuts in a shallow pan in the oven for 5 minutes. Let cool; chop.
Mix sugar, fruit puree fat replacement, cocoa, coffee and oil in a heavy medium saucepan. Bring to a simmer over low heat, whisking constantly. Remove from heat. Add chocolate chips and stir until melted. Transfer to a bowl and let cool for 10 minutes.
Meanwhile, whisk flour, baking soda and salt in another bowl.
Whisk egg, egg white and vanilla into the chocolate mixture. Add half of the flour mixture and stir just until combined. Stir in yogurt. Add remaining flour mixture and toasted nuts, stirring just until combined.
Scrape batter into prepared pan. Bake until a cake tester comes out clean, 25 to 35 minutes. Let cool in the pan on a wire rack.
To make frosting: Mix confectioners' sugar, cocoa, milk and baking soda in a medium saucepan. Whisk constantly over low heat until steaming. Transfer to a bowl and cool to room temperature. Add cream cheese and beat with an electric mixer on medium speed until spreadable, about 5 minutes.
Spread frosting over the top of the cake. Cut into squares.