Preheat the oven to 350 degrees F. Line the large muffin tins with large cupcake liners.

Put the butter, coffee, water, and cocoa in a microwave-safe bowl, cover with plastic wrap, and microwave on HIGH until the butter melts, about 2 minutes. Whisk to combine.

Meanwhile, whisk the flour, granulated sugar, baking powder, salt, and baking soda in a large bowl. Beat the sour cream with the egg and vanilla in a small bowl. Whisk the hot cocoa mixture into the dry ingredients. Stir in the sour cream mixture just to combine; don't overmix. Divide the batter evenly between the prepared muffin tins, filling each one about 3/4 of the way full (about 2/3 cup per cupcake). Bake cupcakes until a toothpick inserted in the center of the cake comes out clean, about 30 minutes.

Cool the cupcakes completely in the tin on a rack. Remove from liners and return cakes to a rack.

For the glaze: Put chocolate in a medium bowl. Bring coconut milk and corn syrup to a boil, stirring constantly, then pour it over the chocolate. Let the mixture stand for 10 minutes, then stir until smooth. Spread 1 tablespoon warm glaze evenly over each cake, then scatter the coconut over the top.

Cook's Note We prefer Dutch-process cocoa here over natural cocoa for its fuller, deeper flavor.