Preheat oven to 300 degrees F (150 degrees C). Scatter kale over a large baking sheet.
Place kale in the preheated oven until dried, about 10 minutes; remove to cool completely, about 10 minutes.
Whisk all-purpose flour, whole wheat pastry flour, caraway seeds, baking powder, baking soda, and salt together in a bowl.
Beat butter, goat cheese, and sugar together in a bowl until creamy; beat in eggs. Stir kale and onion into butter mixture; stir in flour mixture until a soft dough forms. Gather dough into a ball, wrap in plastic wrap, and chill in the refrigerator for 30 minutes.
Preheat oven to 375 degrees F (190 degrees C). Line baking sheets with parchment paper.
Roll dough out on a lightly-floured surface 1/2-inch thick; cut into rounds with a small cookie cutter. Place rounds on prepared baking sheets about 1 inch apart.
Bake in the preheated oven until lightly browned, about 10 minutes.