Preheat your oven to 325°F. Generously spray an 8x8-inch baking pan with cooking spray (optional: line the pan with parchment paper to easily remove the cooled cake from the pan later to cut it; spray the parchment with cooking spray, as well).

In a medium saucepan over medium-high heat, bring apple cider to a boil. Reduce heat; simmer, stirring occasionally, for 10 to 15 minutes or until cider is reduced by half (about 3/4 cup). Pour 1/4 cup cider in a bowl or glass measuring cup; set aside. Pour remaining 1/2 cup cider into a small bowl; cool 5 minutes, then whisk in sour cream and vanilla extract. Set aside.

In that same saucepan, melt butter over medium heat. Once melted, continue to cook, stirring constantly, for 2 to 3 minutes or until butter begins to smell nutty, foam up, then turn a deep amber color with brown flecks throughout. Remove from heat; set aside to cool slightly.

In a medium bowl, whisk together flour, baking powder, baking soda, 1/2 teaspoon salt, 1/2 teaspoon cinnamon and 1/4 teaspoon nutmeg until well-combined. Set aside.

In a large bowl, vigorously whisk eggs and 3/4 cup sugar for about 2 minutes or until mixture is a pale yellow, frothy and thickened. Pour 1 tablespoon brown butter into a small bowl (reserve for later), then pour remaining 8 tablespoons butter into egg mixture, whisking constantly, until smooth.

Whisk flour mixture into egg mixture in three additions, alternating with cider-sour cream mixture, beginning and ending with flour mixture. Whisk just until batter is combined.

Pour and spread batter into prepared pan. Bake for 38 to 43 minutes or until cake is deeply golden brown and a toothpick inserted in the center comes out clean. Transfer cake in pan to a cooling rack; poke top of cake all over with a toothpick or cake tester, then spoon 3 tablespoons reserved cider over top of hot cake. Cool 10 minutes.

In a small bowl, stir together remaining pinch salt, 1/4 cup sugar, 1/2 teaspoon cinnamon and 1/4 teaspoon nutmeg. Whisk together remaining 1 tablespoon cider with remaining 1 tablespoon butter until combined (if needed, heat mixture in microwave for a few seconds to re-melt butter). Brush mixture evenly over cake, then sprinkle generously with sugar mixture. Cool completely before slicing.

Store leftover cake tightly covered at room temperature for up to 4 days.