Preheat the oven to 350°F (175°C). Grease and line a 9-inch (22 cm) round cake pan with parchment paper.
Peel and core the apples. Slice them very thinly using a mandoline or sharp knife.
In a bowl, whisk the eggs and sugar until slightly pale, about 3-4 minutes.
Slowly whisk in the vanilla extract, milk, and cooled brown butter.
Fold in the flour, baking powder, salt, and nutmeg until just combined.
Add the sliced apples and gently toss until every slice is coated.
Arrange the apples in horizontal layers inside the prepared pan, pouring any remaining batter between the layers.
Press down gently every few layers to remove air pockets.
Pour any remaining batter over the top.
Bake for 60-65 minutes until deeply golden and fully set.
Cover loosely with foil during the last 15 minutes if browning too quickly.
Remove from the oven and let cool completely in the pan.
Dust with powdered sugar before serving if desired.