Preheat oven to 350°F (190°C). Lightly coat 8-inch (20 cm) metal baking pan with nonstick cooking spray or oil; set aside.
In a large bowl, combine flour, cinnamon, baking powder, baking soda, salt, nutmeg and ginger; set aside.
In another large bowl, using an electric mixer on high-speed, beat butter and brown sugar until light and fluffy. Add eggs, one at a time, mixing lightly to incorporate after each one. Add applesauce, molasses and vanilla. Beat on medium speed until combined. Pour wet ingredients over dry ingredients and using a wooden spoon or spatula, mix until combined. Spread into prepared pan.
Bake for 30 to 35 minutes or until a toothpick inserted in centre comes out clean. Let cool in pan on rack.
Meanwhile, in a large bowl, using an electric mixer on high speed, beat cream cheese and butter until smooth. Beat in vanilla and then icing sugar, one-third at a time, until smooth. Add more icing sugar as needed until frosting reaches desired consistency.
Remove cake from pan. Once cool, spread frosting over top and sides of cake. Cut into 12 squares. Serve garnished with a sprinkle of nutmeg or walnuts, if desired.
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