Combine the flour (for dough), granulated sugar (for dough), and salt in a mixing bowl. Cut in the cold cubed butter until the mixture resembles coarse crumbs. Add the egg yolk and drizzle in enough of the 2 tbsp ice water for the dough to just come together when pressed. Form into a 1-inch-thick disk, wrap, and chill 30 minutes.
For the compote, cook the chopped apples (for compote), butter (for compote), granulated sugar (for compote), lemon juice (for compote), and vanilla in a covered saucepan over medium heat, stirring occasionally, until the apples break down and the mixture is thick and jammy, 12–15 minutes. Mash to a coarse puree and let cool completely.
Lightly flour the work surface with the flour for dusting. Roll the chilled dough into a 12-inch round and fit it into a 9-inch tart pan with a removable bottom; trim the edges and dock the base with a fork. Chill 10 minutes. Preheat the oven to 375°F.
Line the shell with parchment and fill with pie weights. Bake 15 minutes, then remove the weights and parchment and bake 5–7 minutes more, until the base looks dry and the edges just begin to color. Cool slightly.
Spread the apple compote evenly in the shell. Toss the sliced apples (for topping) with the lemon juice (for apples), then arrange the slices in concentric circles over the compote. Brush the apples with the melted butter (for brushing apples) and sprinkle the granulated sugar (for topping) evenly over the top.
Bake at 375°F until the apples are tender and lightly caramelized at the edges and the crust is deep golden, 35–40 minutes, rotating the pan halfway. If the top browns too quickly, tent loosely with foil.
Warm the apricot jam with the water (for glaze) over low heat until fluid, then brush it over the hot tart. Cool 15–20 minutes, release the tart ring, and serve warm or at room temperature.