Preheat the oven to 350 F, and position a rack in the center of the oven. Line the bottoms of two 9-inch round cake pans with circles of parchment paper. Lightly spray the paper with non-stick baking spray.
Using a small food processor or ninja, grind the oats into fine crumbs resembling oat flour. In a large bowl, whisk together the ground oats with the cake flour, granulated sugar, baking powder, salt, cinnamon and ginger. Set aside.
In a saucepan over medium heat, melt the butter. Continue to cook the butter, swirling the pan occasionally, until nutty brown solids form at the bottom of the pan. This will happen right about the same time that the butter stops hissing and a layer of foam forms on top. Set the butter aside to cool for 10 minutes.
In a bowl, whisk together the browned butter with the brown sugar, oil and molasses, until well combined. Whisk in the eggs, milk and vanilla.
Add the wet ingredients to the dry, and whisk just until combined.
Divide the batter between the prepared pans. Bake the cakes for about 25-30 minutes, until a toothpick or cake tester inserted in the center of the cakes comes out clean, or with moist crumbs clinging to it. Do not over-bake these cakes, or they'll end up dry and crumbly due to the addition of the oats.
Set the pans on a cooling rack, cover them loosely with a clean kitchen towel, and cool completely before frosting.