In a medium saucepan, cook butter over medium-high heat until browned. Immediately transfer browned butter to a small bowl and set aside. Using 2 Tbsp. browned butter, grease donut pan or muffin tin to make 12 donuts or muffins. Preheat the oven to 400°F. Place the flour, flax seed, baking powder, baking soda, nutmeg, ginger, salt, brown sugar, 1/2 tsp. xanthan gum and 1/2 tsp. cinnamon in a large bowl. Whisk well and set aside In a small bowl, combine 1/4 tsp. xanthan gum and 1/2 cup cider; add 2 Tbsp. brown butter, egg and vanilla. Pour liquid mixture into flour mixture, stirring until just combined. Fold in apple. Divide batter evenly into prepared pan and bake for 7-10 minutes or until golden brown and a toothpick inserted in the center comes out clean. A longer cooking time will be necessary for muffins. Set aside to cool for 5 minutes. In a small bowl, whisk together the remaining brown butter and apple cider. Set aside. In another small bowl, combine remaining cinnamon with the sugar. Set aside. Once the donuts are cool enough to touch, remove them from the pan and immediately dip them in the brown butter mixture, then the cinnamon sugar mixture to coat. Serve warm. Donuts are best eaten the day they’re made but may be kept in an airtight container at room temperature up to 2 days.