Pulse the flour, sugar, and cold butter together until the mixture looks like coarse crumbs. Add the egg yolk and just enough ice water for the dough to come together into a ball. Wrap it in plastic and chill it in the fridge for at least 30 minutes before rolling.
Roll the chilled dough out on a lightly floured surface and fit it into your tart pan, pressing it gently into the edges. Prick the base with a fork, line it with parchment and pie weights, and bake at 375°F for about 15 minutes until lightly golden. Let it cool slightly before filling.
While the crust bakes, cream the softened butter and sugar together until fluffy. Beat in the eggs one at a time, then stir in the almond flour, extra flour, and almond extract. You'll end up with a thick, spreadable almond cream that smells incredible already.
Toss your peeled, cored pear halves with lemon juice to stop them from browning. Slice each half thinly, keeping the slices together so you can fan them out. This little trick makes the tart look far more polished than the effort it actually takes.
Spread the almond frangipane evenly over the cooled tart shell. Gently fan out the sliced pear halves and arrange them on top in a circular, spoke-like pattern, pressing lightly so they sit into the cream.
Bake at 375°F for about 35 to 40 minutes, until the frangipane is puffed, golden brown, and springs back when touched. The pears should look soft and slightly caramelized around the edges.
Warm the apricot jam slightly and brush it over the top of the tart while it's still warm. This step gives it that beautiful glossy, bakery-style finish. Let the tart cool before slicing so it holds together properly.