In a large bowl, mix the flour, sugar, and salt. Add the cold, cubed butter and use a pastry blender or your fingertips to cut the butter into the flour until the mixture resembles coarse crumbs. Gradually add the ice water, one tablespoon at a time, mixing until the dough just comes together. Form the dough into a disk, wrap it in plastic wrap, and chill in the refrigerator for at least 30 minutes.

On a lightly floured surface, roll out the chilled dough into a 12-inch circle. Carefully transfer the dough to a 9-inch tart pan with a removable bottom. Gently press the dough into the pan, trimming any excess from the edges. Prick the bottom of the crust with a fork to prevent it from puffing up during baking.

In a large bowl, cream together the softened butter and sugar until light and fluffy. Beat in the almond flour until well combined. Add the eggs one at a time, mixing well after each addition. Stir in the almond extract until evenly distributed.

Pour the frangipane filling into the prepared tart crust, spreading it evenly. Arrange the apricot halves, cut-side up, on top of the frangipane filling.

Preheat your oven to 375°F (190°C). Bake the tart for 35-40 minutes, or until the crust is golden brown and the frangipane filling is set and lightly golden. If the crust starts to brown too quickly, you can tent it with foil during the last 10-15 minutes of baking.

While the tart is baking, prepare the apricot glaze by heating the apricot jam and water in a small saucepan over low heat. Stir until smooth and combined. Once the tart is out of the oven, brush the warm glaze over the apricots to give them a glossy finish.

Let the Apricot Frangipane Tart cool completely in the pan before removing it. This will help the filling set properly. Before serving, garnish with sliced almonds, if desired.