Preheat air fryer to 320 degrees F (160 degrees C). Coat a 6-inch cake pan with cooking spray. Line bottom of pan with parchment paper; dust with 2 tablespoons flour, tapping out excess.

Stir together 1/4 cup flour, the cocoa, baking powder, baking soda, and salt in a small bowl. In another small bowl, whisk together buttermilk, applesauce, oil, espresso powder, and vanilla.

In a third bowl, beat egg with an electric mixer at low speed. Gradually add sugar. Increase mixer speed to high and continue beating until mixture is thick and pale, about 2 minutes.

Alternately fold flour mixture and buttermilk mixture into egg mixture with a rubber spatula, stirring until incorporated before adding next addition. Fold in dark chocolate.

Pour batter into the prepared pan. Arrange pan in the air fryer basket.

Cook until a toothpick inserted in center comes out clean, about 25 minutes. Let cool in pan on a wire rack about 10 minutes. Remove cake from pan and allow to cool completely on a wire rack.

For the frosting, melt butter in a heavy saucepan over medium heat; cook, stirring occasionally, until butter begins to turn golden brown and smells nutty, 15 to 20 minutes. Transfer butter to a bowl; cool for 10 minutes.

Whisk in honey and vanilla. Whisk in powdered sugar until smooth, adding water, 1 teaspoon at a time, until desired consistency. The frosting will look separated but will come together with whisking in water a little at a time.

Immediately top cooled cake with the frosting and garnish with additional dark chocolate.