Preheat oven to 350 degrees. Line two muffin tins with paper liners. In a large bowl or the bowl of a stand mixer, whisk together flours, sugar, baking powder, and 1 tsp. salt. Add 1 cup cold butter, cut into small cubes, and mix until the mixture resembles coarse crumbs. Add the eggs one at a time, beating well after each addition. Combine 1 1/2 cups milk and 2 tsp. vanilla in a measuring cup, and add about half of the milk mixture to the flour mixture. Beat until well blended and smooth. Add the other half of the milk mixture, and beat until smooth. Fold in chocolate chips. Fill paper liners 2/3 full and bake 20 minutes or until a toothpick inserted into the center comes out clean. Let cool in pan for a few minutes and then remove to a cooling rack to cool completely.
In a large bowl, combine the butter and brown sugar. Beat until light and fluffy. Add 1/2 tsp. salt, 2 Tbsp. milk, and vanilla and beat to combine. Add the powdered sugar and beat until light and fluffy. Pipe onto cooled cupcakes and sprinkle with mini chocolate chips.