In a medium saucepan, brown butter over medium-high heat until it smells nutty and fragrant and is deeply browned. Transfer to a small bowl to cool. Set aside.
In a large bowl, whisk flour, flax seed, baking powder, baking soda, ½ teaspoon cinnamon, nutmeg, ginger, brown sugar, Set aside. In a small bowl, combine two tablespoons brown butter, egg, vanilla, and buttermilk. Pour the wet ingredients into dry stirring until just combined. Fold in chopped apple.
Grease two, six portion donut pans or a 12-cup muffin pan with two tablespoons of cooled browned butter. Divide batter evenly into pans and bake in a 400°F oven for 7 to 10 minutes or until golden brown and a toothpick inserted into center comes out clean. (Adjust cooking time for muffins; may need more time..) Remove from oven and let cool five minutes.
In a small bowl, whisk together the remaining four tablespoons of brown butter and apple cider. In another small bowl, whisk together two tsp. cinnamon and granulated sugar. Set aside.
Once the donuts are cool enough to touch, remove them from the pan and immediately dip in butter/cider mixture, then coat in cinnamon sugar. Serve warm. Donuts are best eaten the day they’re made but can be kept in an airtight container at room temperature for up to two days.