Step 1: Pulse the flour, 1/4 cup sugar, and cold butter in a food processor until the mixture resembles coarse crumbs. Add the egg yolk and ice water, pulsing until a dough just begins to form.
Step 2: Wrap the dough in plastic and chill for at least 30 minutes. Once chilled, roll it out and press it into a 9-inch tart tin. Chill the shell again for 15 minutes while preparing the filling.
Step 3: Poach the pear halves in water with a bit of sugar and vanilla for 10 minutes until slightly tender. Drain and pat dry.
Step 4: In a bowl, cream together the softened butter and 1/2 cup sugar. Stir in the almond flour, cardamom, and vanilla to create the frangipane cream.
Step 5: Spread the frangipane cream evenly over the bottom of the chilled tart shell.
Step 6: Slice the poached pear halves thinly, keeping the slices together. Fan them out and arrange them on top of the frangipane.
Step 7: Bake at 375°F (190°C) for 35-40 minutes, or until the pastry is golden and the almond filling is puffed and set.