Whisk the flour, apple pie spice, baking powder, baking soda and salt in a medium bowl; set aside. Beat the butter, brown sugar and granulated sugar in a large bowl with a mixer on medium speed until light and fluffy, about 3 minutes. Reduce the mixer speed to low; beat in the eggs one at a time, then beat in the vanilla. Add the flour mixture and beat until just incorporated. Stir in the oats and cranberries with a wooden spoon. Cover with plastic wrap and refrigerate until firm, at least 1 hour or overnight.
Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Form the dough into 2-inch balls and arrange 2 inches apart on the prepared baking sheets; press with a spatula to flatten. Bake, switching the pans halfway through, until the cookies are golden but the centers are still soft, 15 to 20 minutes. Slide the cookies from the parchment onto racks to cool completely.
Make the glaze: Whisk the confectioners' sugar and milk in a medium bowl. Drizzle on the cookies; let set 5 minutes.