Whisk together all flours and salt in a medium bowl. Set aside.
In a large bowl of an electric mixer, beat butter on medium speed until creamy. Add sugars and vanilla. Beat until light and fluffy, approximately 5 minutes.
Add dry ingredients and mix on low speed until well-blended.
Form dough into two disks. Wrap in plastic wrap and chill for 45 minutes or up to one week.
Preheat oven to 350° F.
On a lightly floured surface, roll out dough 1⁄4 inch thick. Using cookie cutters dipped in flour, cut out cookies. Place on a large cookie sheet lined with parchment paper. Chill 10 minutes before baking.
Bake chilled cookies 10 to 12 minutes or until edges begin to brown slightly. The centers should remain light in color. Cool completely on wire racks.
In a small bowl, whisk cream into powdered sugar until sugar dissolves and thickens. Continue adding more cream a teaspoon at a time to achieve desired spreading consistency.