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Prepare the tart shell: Place a 9 or 10 x 1‑inch tart pan with a removable bottom on a baking sheet. 

On a lightly floured surface using a lightly floured rolling pin, roll pastry dough to about a 12 or 13-inch round, a generous 1/8‑inch thick. Carefully fold the dough in half, and slide it onto a rolling pin. Transfer to a 9 or 10-inch fluted tart pan with removable bottom. Unfold the dough, easing it gently into the pan; do not stretch the dough. Press the dough into place, then run your roller over the edges of the pan, trimming it flush with the top all the way around. Chill in the freezer until firm, about 20 minutes.

Meanwhile, set the oven rack to the center position and preheat to 375 degrees F.

Once chilled, prick the bottom of tart shell all over with a fork, line with parchment paper that extends beyond the rim at least 1 ½‑inches all around. Fill with pie weights or dried beans; bake on the center rack of the heated oven for 15 minutes. Remove parchment and weights, and continue to bake until set and lightly golden, about 8–10 minutes more. Remove the tart pan from the oven one last time and brush with egg whites to seal the holes. Bake for 5 minutes more. Let the tart shell come to room temperature before continuing.

Make the filling: Whisk together the eggs and sugar until pale and frothy. Use a rubber spatula to fold in flour and salt until well-combined, lump-free, but not yet stiff. Set aside. 

In a small saucepan, heat the butter until foamy. Continue heating, swirling the pan often, until bubbles subside, the butter is brown and gives off a nutty aroma. Remove from heat, allow to cool slightly, and stir in vanilla paste. 

Whisking continuously, slowly drizzle the warm (not hot) brown butter vanilla mixture into the egg mixture in a slow, steady stream until the batter is well-combined. Set aside.

Bake the tart: Preheat the oven to 350 degrees F. Place the prepared tart shell on a baking sheet and spread the peeled and diced persimmon evenly across the bottom. There should be some space between the persimmon, don’t use it all if there is not. Pour just enough of the brown butter batter over the persimmon to just cover. It’s fine if edges of the persimmon peek out of the batter. Do not overfill the tart as it rises during baking. If you don’t use all the batter you may bake it separately in a small ramekin. 

Carefully move the tart on its baking sheet to the heated oven and bake for 50 to 55 minutes. When cooked, the top will form a crisp layer that separates from the rest of the filling, but it should be soft in the center. It may require an additional 5 minutes. Use your judgment. The edges of the tart may be rather dark. This is expected. However, if the edges get too dark you may drape them in foil for the last 8 or 10 minutes of baking.

To serve: Allow the tart to cool at least 1 hour before attempting to get the tart out of the shell then slice the tart and dollop with vanilla-scented whipped cream

Whisk egg yolks and cream in a small bowl; set aside. 

Pulse flour, butter, sugar, and salt in a food processor several times until well combined. Add the butter and pulse for 2 or 3 seconds 8–10 more times, until the mixture resembles coarse meal. Remove the lid and drizzle egg-cream mixture all over the flour and butter, and replace the top. blend just to combine (do not overwork dough or crust will be tough). Pulse several more times to distribute moisture.

Transfer dough to a large work surface. Knead just to incorporate, 4–5 turns. Shape the dough into a 6‑inch disc. Wrap the disc in plastic and chill at least 2 hours and up to 24 hours.