Heat the oven to 375ºF with a rack in the middle position. Coat a 9-inch springform pan with butter, dust evenly with flour, then tap out the excess.

In a 12-inch skillet over medium-high, melt the butter. Cook, swirling the pan frequently, until the milk solids at the bottom are golden brown and the butter has a nutty aroma, 1 to 3 minutes.

Pour into a small heatproof bowl without scraping out the skillet. Stir the allspice into the butter and set aside.

Add all of the apples, 2 tablespoons of the sugar and the salt to the still-hot skillet and set over medium-high. Cook, stirring occasionally, until all moisture released by the apples has evaporated and the slices are beginning to brown, 12 to 15 minutes.

Add the brandy and cook until evaporated, 30 to 60 seconds. Transfer to a large plate, spread in an even layer and refrigerate uncovered until cool to the touch, 15 to 20 minutes.

In a small bowl, whisk together the flour and baking powder. In a large bowl, whisk together the eggs, vanilla and 9 tablespoons of the remaining sugar. Gradually whisk in the browned butter.

Add the flour mixture and stir with a rubber spatula until smooth; the batter will be very thick. Add the cooled apples and fold until evenly coated with batter. Transfer to the prepared pan, spread in an even layer and sprinkle with the remaining 1 tablespoon sugar.

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 remove the sides before slicing.