Preheat oven to 350°F. Coat a 9-inch square baking pan with baking spray; set aside.

Bring apple cider and maple syrup to a boil in a medium saucepan over medium-high. Boil, stirring occasionally, until reduced to ½ cup, 12 to 15 minutes. Pour 2 tablespoons cider–maple syrup mixture into a small bowl; set aside for frosting. Let remaining 6 tablespoons cider–maple syrup mixture cool, about 10 minutes.

Meanwhile, beat brown sugar and butter in a large bowl with an electric mixer on medium speed until light and fluffy, 3 to 4 minutes. With mixer running on medium speed, add eggs, 1 at a time, beating until just combined after each addition, about 30 seconds total. Add yogurt and vanilla; beat on medium speed until combined, about 20 seconds. Whisk together flour, baking powder, baking soda, cinnamon, nutmeg, salt, and cloves in a medium bowl until combined. With mixer running on low speed, gradually add flour mixture, beating until just a few streaks remain, about 30 seconds. Add chopped apple and reserved 6 tablespoons cider–maple syrup mixture; beat on low speed just until combined, about 30 seconds. (Batter will be quite thick.)

Scrape batter into prepared baking pan, and spread into an even layer. Bake until a wooden pick inserted into center of cake comes out with just a few moist crumbs, 30 to 35 minutes. Let cool completely in baking pan on a wire rack, about 2 hours.

Beat cream cheese and butter with an electric mixer on medium speed until creamy, about 30 seconds. Add salt and 1 tablespoon reserved cider–maple syrup mixture; beat on low speed until combined, about 1 minute. With mixer running on medium speed, gradually add powdered sugar, beating until smooth and fluffy, 1 to 2 minutes, stopping to scrape down sides of bowl as needed.

Spread frosting over cooled cake. Drizzle with remaining 1 tablespoon reserved cider–maple syrup mixture; garnish with additional cinnamon, if using. Cut into slices, and serve.