Preheat oven to 400° F.
Combine flours, salt, and sugar. Mix well.
Add softened butter and with a pastry blender or two forks, mash to combine.
Take 1 cup of the corn and pulse it in a food processor until smooth. Empty contents of the food processor into the butter/flour bowl, then add the remaining 1/2 cup corn kernels into the same bowl. Mix to combine.
Measure out 1/3 cup of corn cake dough, and pat the dough into a disk shape, about 3/4 inch tall. Lay 6 of the corn cakes on a baking sheet lined with parchment paper.
Bake for 20-25 minutes, flipping over halfway through baking.