Make the Crust: In a large bowl, whisk together the flour and salt. Cut in the cold butter using a pastry blender or your fingers until the mixture resembles coarse crumbs. Gradually add the ice water, mixing until the dough just comes together. Form the dough into a disc, wrap it in plastic wrap, and chill in the refrigerator for at least 30 minutes.

Roll Out the Dough: 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. Press the dough into the bottom and up the sides of the pan. Trim any excess dough and crimp the edges for a decorative finish.

Prepare the Frangipane Filling: In a large bowl, cream together the softened butter and sugar until light and fluffy. Beat in the eggs one at a time, then stir in the almond extract and almond flour. Mix until everything is well combined.

Assemble the Tart: Pour the frangipane filling into the prepared tart crust, spreading it evenly. Arrange the plum halves on top of the frangipane in a decorative pattern, cut-side up. Sprinkle the plums with 2 tablespoons of granulated sugar and almond flakes.

Bake the Tart: Preheat your oven to 375°F (190°C). Bake the tart for 45-50 minutes, or until the crust is golden brown and the frangipane filling is set. If the crust starts to brown too quickly, you can cover it loosely with foil during the last 15 minutes of baking.

Glaze the Tart (Optional): While the tart is cooling, heat a few tablespoons of apricot jam in a small saucepan or in the microwave until melted. Brush the warm jam over the plums to give them a beautiful shine.

Cool and Serve: Let the Plum Frangipane Tart cool completely in the tart pan before removing it. Once cooled, carefully remove the tart from the pan and transfer it to a serving plate.