To make the banana cake doughnuts: Preheat the oven to 350°F. Lightly grease three 6-mold doughnut pans with butter. Set a wire rack in a parchment-lined rimmed baking sheet and set aside.
In a small saucepan, cook the butter over low heat until the butter is caramel colored and smells toasty.
Strain the butter into the bowl of a stand mixer. Using the paddle attachment, beat together the butter, sugar, honey, and vanilla until it is cooled off slightly. Add the eggs one at a time, mixing well after each. Add the banana puree and mix until incorporated.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, and cinnamon. Set aside.
Alternate adding the flour mixture and buttermilk to the butter mixture, starting and ending with the dry ingredients.
Use a pastry bag fitted with a round tip to put the batter into the prepared doughnut molds, filling each about two-thirds full. Bake until a tester comes out clean, about 15 minutes. Remove from the pan.
To make the honey glaze: In a medium bowl, stir together the sugar, heavy cream, and honey until smooth. Dip one side of each hot doughnut into the glaze, let cool,on the prepared baking sheet, and dip again.