Begin by reducing the apple cider. Add apple cider to a medium-sized saucepan and bring it to a boil. Once it is boiling, reduce the heat to low and allow it to simmer for 20–30 minutes, stirring occasionally. Once the cider has reduced, you should have ½ a cup of cider. Set aside, or in the refrigerator, to cool.
Preheat the oven to 375°F. Grease an 8x8 baking pan and line with parchment paper. Set aside while you make the cake batter and crumb topping.
Place the butter in a small saucepan over medium-high heat. Stir for 5–7 minutes or until you have a foamy, golden, nutty butter. This is also referred to as "browned butter". Transfer to a small bowl, and set aside to cool while you work on the streusel crumb.
In a small bowl, combine brown sugar, flour, cinnamon, salt and melted butter. Mix with a pastry blender, or fork, until you have a coarse crumb and set aside.
In a bowl, combine the "dry ingredients". The flour, cinnamon, allspice, nutmeg, ginger, salt, baking soda, and baking powder. Use a whisk to combine.
In another medium-sized bowl, combine the "wet ingredients". The slightly cooled brown butter, brown sugar, eggs, sour cream, apple sauce and vanilla. Mix until combined.
Add the dry ingredients to the wet ingredients. Mix until partially combined. Add the reduced apple cider. Mix again until well incorporated, being careful not to over mix the batter.
To your prepared cake pan, add half of your cake batter to the pan, spread evenly. Sprinkle with half the crumb mixture. Add the remaining cake batter and finish the top with the remaining crumb streusel. Bake for 35–45 minutes in a preheated oven.
While the cake bakes, work on the glaze by combing the confectioners' sugar, milk, and vanilla bean paste. Mix until well incorporated. Set aside.
Remove the cake and cool for 30 minutes. Drizzle with the glaze and enjoy! Store extra cake in an airtight container or freeze to use later.