Preheat oven to 350°F (175°C). Grease an 8×8-inch baking dish with nonstick spray.
In a medium bowl, whisk together flour, baking soda, ginger, cinnamon, and pumpkin pie spice.
In a small saucepan, warm the milk and brown sugar just until the sugar melts. Remove from heat.
In another saucepan, heat molasses and corn syrup until lukewarm.
Cut the cold butter into the dry ingredients using a pastry blender until crumbly.
Pour the warm milk mixture into the flour mixture and stir just until combined.
Add the molasses mixture and mix until smooth.
Whisk in the egg until fully incorporated.
Transfer batter to the prepared baking dish and bake for 45 minutes, or until a toothpick comes out clean.
Allow the cake to cool completely before frosting.