Apple Tart

Make the pastry

1. Place the flour, sugar, butter, and salt in a food processor and pulse until the mixture is just combined (somewhere between 5 and 10 pulses). Add 4 tablespoons water and pulse just until the dough holds together. If necessary, add more water, 1 tablespoon at a time, and pulse just until the dough begins to clump. Wrap the dough in waxed paper and refrigerate for 30 to 60 minutes.
2. Let the dough rest at room temperature for about 10 minutes. Butter and lightly flour an 11- or 10-inch round tart pan with a removable bottom.
3. Roll out the pastry dough on a lightly floured work surface to a circle slightly larger than the prepared tart pan. Transfer the dough to the pan, gently pressing the dough into the edges, and trim the edges by running a rolling pin over the top of the tart pan, using the edge of the pan as a cutting surface. Let the excess pastry fall away. Refrigerate for at least 30 minutes, until firm.
4. Preheat the oven to 400ºF (200ºC).
5. Prick the pastry base all over with the tines of a fork. Line the pastry with parchment paper and fill it with baking weights or dried beans. Bake for 15 minutes, then remove the parchment and weights, and bake for 10 to 15 minutes more, until golden. Let cool to room temperature.

Make the apple purée

1. Place the diced apples, vanilla pod, sugar, butter, and 3 tablespoons water in a saucepan over medium heat. Cook gently, stirring often, for 10 to 15 minutes, until the apples are softened, adding more water if necessary to keep the apples from scorching.
2. Remove the vanilla pod and use the tip of a small sharp knife to scrape the vanilla seeds directly into the mixture. Discard the pod. Transfer the mixture to a food processor and purée until smooth.
3. Preheat the oven to 375ºF (190ºC).

Assemble the apple tart

1. Spread the apple purée evenly over the pastry. Peel, core, and slice the apples 1/8 to 1/4 inch thick. Arrange the apple slices in a neat circle around the edge; they should be slightly overlapping. Repeat to create an inner circle, trimming the slices slightly so that they fit, going in the opposite direction from the outer circle. Brush the apples with the melted butter and sprinkle with sugar.
2. Bake for 25 to 45 minutes, until the apples are just tender and very lightly browned. (The dramatic difference in timing depends on the thickness and the type of apples you choose. If the apples haven’t browned by the time they're tender, run the tart under the broiler for a minute or so.)
3. Let the tart cool slightly. Remove the sides of the pan and slice and serve the tart warm with cream or half-and-half.