Special equipment: donut hole or mini muffin pans
Preheat the oven to 400 degrees F.
For the eggplant donuts: Halve the eggplant lengthwise, score diagonally and generously brush with olive oil to coat the flesh. Put on a baking pan cut-side up and roast until soft, about 45 minutes. Set aside.
Lower the oven temperature to 350 degrees F. Grease donut hole or mini muffin pans with butter, then coat with flour.
Combine the flour, baking powder, cinnamon, ginger and salt in a large bowl. In a separate bowl, whisk together the milk, vegetable oil, brown sugar and egg until smooth. Remove the roasted eggplant flesh from the skin and mash. Mix the mashed eggplant into the wet ingredients. Add the dry ingredients to the wet and mix until combined.
Using a small scoop, fill each donut hole mold three-quarters full. Bake until an inserted toothpick comes out clean, 15 to 20 minutes.
For the cinnamon-sugar coating: Put the melted butter in a bowl. Combine the granulated sugar and cinnamon in another bowl.
Remove the donuts from the pan and dip each in the melted butter. Roll in the cinnamon sugar to coat.