Preheat the oven to 375ºF. Line the bottom of a 9-inch springform pan with parchment paper. Grease the bottom and sides lightly with butter.
In a large (12-inch) skillet over medium-high, melt the butter. Cook, swirling the pan frequently, until the milk solids at the bottom are just starting to turn golden brown and the butter smells nutty, 1 to 3 minutes. Pour into a small heatproof bowl, but don’t scrape out the skillet (we want a thin layer of butter in the pan for the apples). Into the bowl with the butter, stir in the allspice or cinnamon (if using) and set aside.
To the skillet over medium-high heat, add the apples, 2 tablespoons sugar and the salt. Cook until all water from the apples has evaporated and they are beginning to brown, about 8-12 minutes, stirring occasionally (careful not to burn).
Add the brandy or rum (if using) and cook until evaporated, 30 to 60 seconds. Transfer to a large plate or quarter sheet pan, spread into an even layer and chill in the fridge until just barely warm or cool, about 15 to 20 minutes (you can pop it in the freezer towards the end to speed it up).
In a small bowl, whisk together the flours and baking powder. In a large bowl, whisk together the eggs, vanilla and remaining 2 tablespoons sugar. Slowly whisk in the browned butter, scraping out every last bit. Stir in the flour mixture with a rubber spatula until smooth (the batter will be thick).
Add the cooled apples and use a rubber spatula to fold until evenly coated with batter. Transfer to the prepared pan, spread into an even layer, and sprinkle with a bit more sugar on top, if you’d like.
Bake until deeply browned, 35 to 40 minutes. Let cool COMPLETELY in the pan on a wire rack, about 2 hours. Run a knife around the inside of the pan and carefully transfer the cake to your serving platter/cake stand.
Slice and serve (at room temp or just barely warm) topped with a dollop of whipped cream, whipped yogurt, or crème fraîche. I also like to drizzle good olive oil and a sprinkle of flaky salt on top.