Is there anything more luscious than a sweet, ripe fig? Their season is so fleeting that if you're a fan, you'll want to make the most of them while they're available. They're lovely in this rich tart, which combines the intense flavor of figs with a vanilla-scented custard. This dessert is best served warm the day it's made.
Lightly butter a 9-inch pie plate. On a lightly floured surface, roll out the sheet of puff pastry to a 10-inch circle. Carefully fit the dough into the pie plate. Trim the excess pastry along the outer edge of the pie plate rim. Refrigerate for 30 minutes.
Place the cream, vanilla bean, and brown sugar in a small saucepan and bring to a simmer over medium heat. Stir until the sugar dissolves. Turn off the heat, and add the cinnamon sticks and nutmeg. Let the mixture cool for 15 minutes, then remove the cinnamon sticks and vanilla bean. Scrape the tiny black seeds from the bean into the cream. Wash and dry the bean, and reserve for another use, if desired.
Place 2 eggs and the egg yolk in a medium-size bowl and whisk to combine. Slowly add the warm cream mixture, whisking constantly.
Position a rack in the lower third of the oven, and preheat to 400 degrees F.
Remove the pastry from the refrigerator and prick the bottom and sides of the dough with a fork. Arrange the fig slices in overlapping concentric circles. Place the tart on a rimmed baking sheet.
Mix the remaining egg with 1 teaspoon of water to make an egg wash. Brush the edge of the puff pastry crust with the wash.
Carefully pour the warm custard cream over the figs. Transfer the baking sheet to the oven. Bake until the custard is set in the middle, 30 to 35 minutes. Transfer the tart to a wire rack and cool for 30 minutes before serving.