Preheat the oven to 350°F. Grease a doughnut pan and set aside
Melt the butter in a small saucepan. When the butter is melted, pour 1/4 cup (this is 4 tablespoons) of it into a bowl and set it aside.
To the saucepan containing the remaining 2 tablespoons of butter, add the cinnamon, brown sugar, milk, cornstarch and vanilla extract. Heat just until it bubbles then take it off the heat and pour around 3/4 tsp of the mixture into the bottom of each cavity of the doughnut pan.
Add the egg and milk to the bowl containing the butter and whisk together.
In a medium bowl, mix together the flour, granulated sugar, baking powder and salt. Pour in the contents of the bowl with the butter mixture into the flour mixture, and mix together until just combined (the mixture may look slightly lumpy, don’t over mix it.) Spoon or pipe the batter into the doughnut pan, filling each cavity about 2/3 full.
Bake in the preheated oven for 8-10 minutes and turn out a wire rack to cool.
Cream the frosting ingredients together in a bowl. Pour into a small sandwich bag and cut the very tip of one of the corners off. Pipe the frosting in zig-zags onto the cooled mini doughnuts.