In a large bowl, whisk together flour, cocoa powder, baking powder, and salt until combined. Whisk in sugar.
Stir in egg, milk, coffee, and oil and mix until combined, but do not overmix. As with muffins, a few lumps are fine.
Fill donut pan wells about 3/4 full. Bake for 12 minutes, or until lightly springy to touch. Remove from pan (a spoon helps here) and let cool on a rack set over a baking sheet. Clean off any stuck bits, re-grease, and repeat with remaining batter.
In a bowl with a bottom wide enough for the donut, whisk together powdered sugar, cocoa powder, milk, and peppermint extract until completely smooth.
Dip donuts into glaze and return to rack. Before the glaze hardens, sprinkle on candy cane bits.