This sleek French-style apple tart gets a nutty depth from a layer of peanut butter frangipane, which grounds thin slices of sweet, crisp apples on a buttery puff pastry base. Dufour puff pastry is best for this tart; look for it in the freezer section of your local grocery store or Whole Foods or online. Use sweet-tart apples such as Pink Lady or Honeycrisp or any other variety you like that will hold its shape while baking. To save time, assemble the tart and freeze it up to 3 days in advance.
Make the peanut butter frangipane: In a small bowl, mash 3 tablespoons brown sugar and the 3 tablespoons room temperature butter together until no lumps of butter remain. Add the peanut butter, ¼ teaspoon salt and 1 egg and stir together until smooth.
Make the apple filling: Cut the apples into quarters, and remove and discard their cores. Using a sharp knife or a mandolin, thinly slice the apples and transfer to a large bowl. Add the remaining 2 tablespoons brown sugar, the lemon juice, vanilla, cinnamon and remaining ¼ teaspoon salt. Toss to combine, then set aside while you shape the tart.
Lightly flour a sheet of parchment paper, then unfold the puff pastry sheet on top. Use a floured rolling pin to gently roll out the pastry, rotating once or twice, until it measures 16 by 12 inches. Slide the parchment paper with pastry onto a rimmed baking sheet. Trim the edges just to straighten them. Cut off a 1-inch strip of dough from each long side of pastry, then repeat on the short sides; set the four strips aside.
In a small bowl, whisk the remaining 1 egg with a splash of water until smooth. Using a pastry brush, brush a 1-inch-strip of egg wash along each side of the dough rectangle. Starting with the two long dough strips, place them over the egg wash on the long sides of the rectangle; repeat with the short sides. Using scissors or a paring knife, trim any overhang from the corners. Brush the new rim you just made with egg wash and sprinkle evenly with the turbinado sugar.
Prick the pastry inside the border all over with the tines of a fork, then scrape the peanut butter frangipane onto the center of the dough. Using an offset spatula or butter knife, spread the frangipane to the edges of the border. Arrange the apple slices in a tiled pattern over the top of the frangipane in whatever fashion you like, then drizzle any juices from the bowl over the fruit. Transfer the baking sheet with the tart to the refrigerator for 20 minutes.
While the tart chills, arrange a rack in the center of the oven and heat the oven to 375 degrees. Cut the remaining 4 tablespoons chilled butter into ¼-inch cubes and arrange them evenly over the top of the apples in the tart. Sprinkle the granulated sugar evenly over the apples as well. Bake the tart until the pastry border is golden brown and puffed and the apples are golden brown and tender in the center, 35 to 40 minutes.
Remove the baking sheet from the oven and transfer to a cooling rack. Let the tart cool for 30 minutes before serving with vanilla ice cream.
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