In a large bowl, whisk together the flour, 1/2 cup granulated sugar, and salt. Add the cold, cubed butter and cut it into the dry ingredients using your fingertips, a pastry blender, or a food processor, until the mixture resembles coarse crumbs with some pea-sized pieces of butter.

In a small bowl, lightly beat the egg yolk with 2 tablespoons of ice water. Add the egg mixture to the flour mixture and mix until just combined. If the dough is too dry, add the remaining 1 tablespoon of ice water, a teaspoon at a time, until the dough comes together. Be careful not to overmix.

Gather the dough into a disc, wrap it tightly in plastic wrap, and refrigerate for at least 30 minutes (or up to 2 days). This chilling step is crucial for a tender, flaky crust.

While the dough chills, peel, core, and thinly slice the apples. Aim for slices about 1/8-inch thick. You can use a mandoline for uniform slices.

In a large bowl, gently toss the apple slices with 1/4 cup granulated sugar, lemon juice, and optional cinnamon. Set aside.

Preheat your oven to 375°F (190°C). Lightly grease a 9-inch tart pan with a removable bottom.

On a lightly floured surface or between two sheets of parchment paper, roll out the chilled dough into a 12-inch circle. Carefully transfer the dough to the prepared tart pan. Press it gently into the bottom and up the sides. Trim any excess dough from the edges. You can patch any tears with excess dough.

Arrange the apple slices decoratively over the pastry crust. You can start from the outer edge and work your way inward, overlapping the slices, or create a spiral pattern. Drizzle the melted butter evenly over the apples and sprinkle with 1 tablespoon of granulated sugar.

Bake for 35-40 minutes, or until the crust is golden brown and the apples are tender. If the crust edges brown too quickly, you can cover them loosely with aluminum foil.

While the tart is still hot, warm the apricot jam and 1 tablespoon of water in a small saucepan over low heat until melted and smooth. Strain the mixture through a fine-mesh sieve if desired, for an extra smooth glaze.

Brush the warm apricot glaze evenly over the baked apples. This adds shine and a touch of sweetness.

Let the tart cool completely on a wire rack before removing it from the pan and slicing. This allows the apples to set and prevents the crust from breaking.

Serve warm or at room temperature, perhaps with a dollop of crème fraîche or vanilla ice cream.