In a large mixing bowl, whisk together flour, sugar, cocoa powders, baking soda and salt.

In a small mixing bowl, stir together beer, oil, vinegar and vanilla extract.

Pour beer mixture over the flour mixture. Whisk until combined and smooth.

Divide the batter evenly among the 12 donut wells. Tap the pans gently on your work surface to dislodge any bubbles.

Bake 15 minutes, or until a toothpick inserted comes out with a few crumbs. Cool in the pans on wire racks for 10 minutes. Unmold the donuts to cool completely on the wire racks.

In a medium saucepan, stir together the confectioner's sugar and espresso powder. Top with Bailey's and water and whisk together. Whisking often, cook the mixture over medium heat until it liquefies. It should be very smooth and warm, but not hot. Do not simmer. Use immediately.

Reduce heat to low and dip the top-side of each donut in the glaze. Place donut right-side-up on a cooling rack. Immediately top with chocolate sprinkles before the glaze hardens.