Preheat oven to 350 degrees. Butter and flour 12 mini Bundt pans. In a medium bowl, whisk together flour, cinnamon, baking powder, salt, and baking soda.
In a large bowl, combine melted butter, 1 1/2 cups brown sugar, and eggs; whisk until smooth. Gradually whisk in flour mixture just until combined (do not overmix). Using a rubber spatula, fold in apples. Divide batter among prepared cavities, using about 1/2 cup per cake. Tap pans firmly on counter to remove air bubbles. Bake until a tester inserted in cake comes out clean, about 30 minutes. Let cool in pan on a wire rack 15 minutes, then invert onto rack.
In a small saucepan, combine cold butter, remaining 1/4 cup brown sugar, and bourbon. Heat over medium, whisking occasionally, until butter is melted and sugar has dissolved, about 3 minutes. Brush syrup over warm cakes. Let cool completely.
Whisk together confectioners' sugar and enough apple cider or water to form a thick yet pourable glaze. Set rack with cakes over a piece of wax paper (for easy cleanup); drizzle each cake 1 to 2 teaspoons glaze and let set before serving. Serve, with dried apple slices.