In a large bowl, combine 2 cups all-purpose flour, 1/4 cup granulated sugar, and a pinch of salt.
Add the cold, cubed butter and work it into the flour mixture with your fingertips or a pastry cutter until it resembles coarse crumbs.
Add the egg and gently knead until the dough comes together into a smooth ball.
Wrap the dough in plastic wrap and refrigerate for 30 minutes.
Preheat the oven to 350°F. Grease and flour a 9-inch tart pan or pie dish.
On a lightly floured surface, roll the chilled dough into a 12-inch circle. Transfer it to the prepared pan, pressing it into the edges, trimming any excess, and prick the bottom with a fork.
Peel, core, and thinly slice the apples.
In a medium bowl, toss the apple slices with 1/4 cup granulated sugar, 1 teaspoon ground cinnamon, and the juice of half a lemon.
Arrange the apple slices in an even layer over the dough and drizzle with the melted butter.
Bake for 40–50 minutes, or until the crust is golden and the apples are tender. If the top browns too quickly, cover loosely with foil.
While the tart bakes, gently heat the apricot preserves with the water in a small saucepan or microwave until melted, then brush the glaze over the warm tart.
Let the tart cool slightly, then remove it from the pan. Serve warm on its own or topped with vanilla ice cream or whipped cream.