In the bowl of a stand mixer fitted with the paddle attachment, or in a medium bowl using a hand mixer, mix the cooled brown butter (from above), brown sugar, Greek yogurt, egg, and vanilla until smooth.
¾ cup light brown sugar, ½ cup Greek yogurt, 1 large egg, 1 teaspoon vanilla bean paste
In a separate small bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon and nutmeg until combined.
1 ¼ cups all-purpose flour, ¾ teaspoon baking powder, ¼ teaspoon baking soda, ¼ teaspoon fine sea salt, 1 teaspoon ground cinnamon, ¼ teaspoon ground nutmeg
Fold the dry ingredients into the wet ingredients until just combined. The cake batter is thick.
Reserve ½ cup of the apples and set aside. Fold the remaining apples into the cake batter.
Scoop the batter into the prepared pan and smooth with an offset spatula. Scatter the remaining reserved apples on top. Use a piping bag or spoon and drizzle the cinnamon sugar topping all over the top of the cake.
Bake for 35-45 minutes. I like to check my cake at 35 minutes. The cake is done when a toothpick comes out with only a few moist crumbs, the center springs back slightly, and the edges are a nice deep golden brown. My cake was finished at 40 minutes.
Remove from the oven and allow the cake to cool in the pan for 20 minutes. Meanwhile, prepare the vanilla glaze.