3/4 cup soft whole-wheat flour
3/4 cup unsifted cake flour
1 tsp baking powder
1/4 tsp salt
2 tbsp butter
1/2 cup sucanat
2 tbsp oil
1 large egg
1 1/2 tsp vanilla extract
Set a rack in the upper third of the oven; preheat to 350°F. Spray two baking sheets with cooking spray.
Whisk whole-wheat flour, cake flour, baking powder and salt in a medium bowl.
Melt butter in a small saucepan over low heat. Cook, swirling the pan, until the butter turns a nutty brown, about 1 minute, and pour into a mixing bowl. Add sucanat and oil; beat with an electric mixer until smooth. Mix in egg and vanilla; beat until smooth. Add the flour mixture and mix on low speed until just combined. Divide the dough in half and press each piece into a disk.
Working with one disk at a time, roll dough on a floured surface to a thickness of 1/8 inch. Cut out cookies with small (about 2- to 2 1/2-inch) cookie cutters. Place the cookies about 1/2 inch apart on the prepared baking sheets.
Bake the cookies in the upper third of the oven, 1 sheet at a time, until slightly golden on the edges, 5 to 7 minutes. Do not overbake. Transfer to wire racks to cool.
Make Ahead Tip: Prepare the dough through Step 3; wrap well and refrigerate for up to 2 days or freeze for up to 1 month. (If frozen, return to room temperature before rolling out.) Store the cookies in an airtight container for up to 3 days or freeze for longer storage.