Preheat oven to 325F. Spray an 9x9 square cake pan with baking spray and line with parchment paper.
Whisk together the flour, oats, and ginger in a medium bowl, then set aside.
Crack the egg into a small bowl, whisk lightly, then set aside.
In a large saucepan over low heat, melt the butter, corn syrup, molasses, and brown sugar, stirring constantly. Adjust heat as necessary to make sure mixture is heated through but does not start bubbling.
Once butter is melted, sugar is dissolved, and all ingredients are thoroughly combined, remove pan from heat.
Stir the dry ingredients into the warm sugar mixture until thoroughly combined.
Add the milk and stir to combine.
Add the beaten egg and stir to combine. **
Pour batter into the prepared cake pan and bake at 350F for 45-50 minutes, until toothpick comes out clean and the top is firm to the touch.
Remove cake from oven and let cool completely, then wrap in plastic wrap and aluminum foil to let sit for 3-4 days.
To serve, remove wrappings and slice into small squares. If you've refrigerated the cake, let it come to room temperature or warm it up before serving.