Preheat oven to 425°. Spray 2 (6-cup) doughnut pans with cooking spray.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter and sugars at medium speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition. Stir in vanilla.

In a medium bowl, whisk together flour, baking powder, salt, cinnamon, and baking soda. With mixer on low speed, gradually add flour mixture to butter mixture alternately with beer, beginning and ending with flour mixture, beating just until combined after each addition. Spoon batter into prepared pans (cups will be full).

Bake until a wooden pick inserted in center comes out clean, 8 to 10 minutes. Let cool in pans for 5 minutes. Remove from pans, and let cool 5 minutes more on a wire rack. Dip doughnuts in Maple-Beer Glaze.

In a medium saucepan, heat maple syrup and beer over medium heat until reduced to ½ cup, 15 to 20 minutes. Pour into a medium bowl. Whisk in cold butter, and let cool for 20 minutes. Whisk in confectioners’ sugar until smooth. Use immediately.