Preheat the oven to 375 F, and position a rack in the center of the oven. Thoroughly grease a standard-sized doughnut pan or several mini donut pans with non-stick baking spray.
In a bowl, whisk together the flour, brown sugar, espresso powder, spices and salt.
Separately, whisk together the pumpkin, sour cream, egg and vanilla.
Add the wet ingredients to the dry, and stir the batter together with a spatula just until combined.
The batter will be thick. Spoon the batter into a piping bag or a plastic bag, and snip off the corner. Pipe the batter into the doughnut pan, filling the molds half way full.
Bake the doughnuts on the center oven rack for about 11-12 minutes for 2 1/2 - 3 inch doughnuts, or about 8-10 minutes for mini doughnuts, until a toothpick comes out clean. Cool the doughnuts in the pan for 1 minute, then turn out onto a wire rack.
Immediately dip the hot doughnuts in the cinnamon/sugar mixture to thoroughly coat them. If you have trouble with the cinnamon/sugar not sticking to the doughnuts, then use a pastry brush to lightly brush a very small amount of melted or soft butter on the doughnuts, and then coat them in the cinnamon/sugar.
Place the doughnuts on a wire rack, and cool for 10 minutes, then serve warm.