Preheat oven to 350˚F. Grease two baked doughnut pans.
To make the cinnamon topping:
In a small saucepan, melt the butter over medium heat. Whisk in the brown sugar, cinnamon, cornstarch, and milk. Once it has almost come to a boil, remove from heat and whisk in vanilla extract.
Pour around 1 tablespoon of the cinnamon topping into each cavity of the greased doughnut pan.
To make the batter:
In a large bowl, combine the flour, sugar, cornstarch, baking powder, baking soda, cinnamon, nutmeg, and salt. Stir in the buttermilk, eggs, vegetable oil, and vanilla extract. Pipe or spoon the batter into the cavities of the pan over the cinnamon topping about 2/3-3/4 full.
Bake in preheated oven until lightly golden and the doughnut springs back with touched, 10-12 minutes. Allow to cool in pan 5-10 minutes before transferring to wire rack to cool to room temperature. A small spoon or knife around the edges will help release them (be careful not to scratch the pan).
To make the glaze:
In a small bowl, whisk the milk into the powdered sugar until smooth. If too thin, add more powdered sugar. If too thick, add a little more milk. Drizzle the glaze over the cooled doughnuts and serve after it sets slightly.