Preheat oven to 400 degrees F (200 degrees C). Line a large baking sheet with parchment paper.
Combine kale and softened butter together in a bowl until a smooth paste forms. Add Cheddar cheese, oat bran, and milk to paste and mix until well blended. Stir all-purpose flour, whole wheat pastry flour, salt, and ginger into paste mixture until dough easily pulls away from the sides of the bowl, adding more flour if dough is sticky.
Chill dough in refrigerator for 30 minutes.
Roll dough to 1/2-inch-thickness on a lightly floured surface. Cut dough into rounds using a cookie cutter or small lid and place on the prepared baking sheet about 2 inches apart.
Bake in the preheated oven until lightly browned, about 10 minutes.