Preheat the oven to 350 degrees. Grease 2 6-cavity donut pan (or 1 12-cavity donut pan) with melted butter or nonstick cooking spray.
In the bowl of a standing mixer, combine sugar, eggs, pumpkin puree. Mix until completely blended.
Melt the butter in a medium sauce pan over medium low heat (use a light colored pan if possible). Continue to heat the melted butter, swirling the pan occasionally, until it starts to brown and you notice a nutty aroma. Once the butter browns keep a close eye on it because it can burn easily at this point. Immediately remove the pan from the heat and pour the browned butter into a bowl or measuring cup so that it can cool.
Add the cooled browned butter to the egg mixture and mix until fully combined.
Add the flour, baking powder, baking soda, salt, and cinnamon to the egg mixture and mix until fully combined.
Use a rubber spatula to scoop the batter into a large, gallon zip-lock bag. Cut off one corner of the bag and pipe the batter into the prepared donut dish. Bake for 18-25 minutes until a toothpick inserted into the donut comes out clean.
Allow the donuts to cool completely before frosting them.
While the donuts cool, combine the dark chocolate and half and half in a heat proof bowl. Bring 2 inches of water to a simmer in a small saucepan. Keep the heat on low and place the bowl of chocolate over the pot of simmering water.
Stirring constantly, allow the chocolate and half and half to melt together until completely smooth. Remove from the heat.
Dip the cooled donuts into the melted chocolate and smooth with a butter knife. Let the chocolate harden at room temperature.