Prepare the pan. Preheat the oven to 350°F (175°C). Grease a 9-inch springform pan and line the base with parchment paper.
Brown the butter. Melt the unsalted butter in a small saucepan over medium heat, stirring frequently. Continue cooking as it foams until the milk solids turn golden brown and smell nutty. Immediately scrape into a heatproof bowl. Stir and measure out 95 g, including the browned bits. Let cool until warm, not hot.
Prepare the apples. Peel, core, and chop enough Gala and Honeycrisp apples to give you 450 g of ½-inch chunks. Slice the additional apple about ⅛ inch (3 mm) thick for the topping.
Mix the dry ingredients. Whisk the flour, baking powder, cinnamon, and salt together in a medium bowl.
Mix the wet ingredients. In a larger bowl, whisk the eggs and ¾ cup sugar for about 1 minute, until slightly pale and foamy. Whisk in the vanilla, milk, and cooled brown butter.
Combine. Gently fold in the dry ingredients just until no dry flour remains, then fold in the apple chunks.
Add the topping. Spread the batter evenly in the pan. Arrange the apple slices in a lightly overlapping circle or fan, keeping to one layer and leaving some batter visible. Sprinkle with the remaining tablespoon of sugar.
Bake. Bake on the middle oven rack for 45–55 minutes, or until the centre is set and a skewer inserted into cake, avoiding apple pieces, comes out without wet batter. Continue in 5-minute increments if needed. Loosely cover with foil if the top browns before the centre finishes.
Optional broiled finish. Once fully baked, broil for 30–60 seconds if you would like more colour on the apple topping. Watch continuously and remove as soon as it is golden. Ensure the pan is broiler-safe, and no exposed parchment is near the element.
Cool. Let cool in the pan for 20 minutes. Run a thin knife around the edge and release the springform ring. Cool for at least another 20 minutes before slicing, or cool completely for cleaner slices.