Pulse the flour, sugar, and salt in a food processor. Then add the cold butter and pulse until the mixture looks like pea-sized crumbs. Add the ice water one tablespoon at a time, pulsing just until the dough comes together. Shape it into a disc, wrap it, and refrigerate for 30 minutes.

Preheat the oven to 375°F (190°C). Roll the chilled dough on a floured surface into a 12-inch circle, then fit it into a 9-inch tart pan with a removable bottom.

Press it into the fluted edges, trim the extra dough, and prick the bottom with a fork. Line the crust with parchment paper, fill it with pie weights or dried beans, and blind bake for 15 minutes.

Remove the weights and parchment, then bake for another 5 minutes until pale golden. Let it cool slightly.

Beat the almond flour, sugar, and softened butter together until smooth. Next, beat in the eggs one at a time.

Then add the flour, almond extract, and salt, and continue mixing until the filling is light and creamy. Spread the frangipane evenly into the warm par-baked crust.

Thinly slice each pear half crosswise, but keep the slices together. Then fan them out slightly with your hand.

Arrange the fanned pear halves on top of the frangipane in a circular pattern, cut-side down, and press them gently into the almond cream.

Bake the tart for 35 to 40 minutes, or until the frangipane is puffed, golden brown, and a toothpick inserted in the center comes out clean.

Let the tart cool for 15 minutes. Then brush the warm top with the warmed apricot preserves for a glossy finish and scatter the sliced almonds around the edge. Serve warm or at room temperature.