There's so much to love about this tart, starrting with the super simple but totally delicious press-in crust. While it calls for chestnut flour, you can substitute with more almond flour. Then there's the filling. Guess what--it's apples sprinkled with sugar. That's it! Yes with or without the finishing caramel sauce, it's stunning to the eye and the taste buds. If you’re short on time or crave a more traditional finish, skip the caramel and brush the tart with a few tablespoons of apricot jam loosened with a few drops of water, heated, and strained. Best the day it's baked, the tart will keep at room temperature for 1 day but leftovers may be stored in the fridge for up to 2 days.
In the bowl of a stand mixer fitted with the paddle attachment, combine the almond, sweet rice, and chestnut flours with the tapioca flour, sugar, and salt. Scatter the butter pieces over the top and drizzle with the vanilla. Turn the mixer to medium-low and run until the butter is completely integrated, 3 to 5 minutes. (If the dough doesn’t come together, work it with your hands.)
Cut the flesh off each apple core in large chunks, cutting close to the core (alternatively, halve and core the apples). Cut each chunk into slices as thin as possible, about 1⁄8-inch thick, keeping the slices stacked together and discarding the rounded edges. Fan the slices out lengthwise slightly and place them, cut-side facing the center of the tart, around the outside of the shell, tucking the last apple under the first to make a continuous loop. Repeat with a second layer just inside the first, reversing the fan if you like, and continue until you’ve filled the tart, tucking in as many of the extra pieces as you can; the apples will reduce a lot as they cook. Sprinkle all over with the sugar and dot with the butter.