Prepare the Dough: In a food processor, pulse the flour, sugar, and salt. Add the cold butter cubes and pulse until the mixture resembles coarse meal with some pea-sized butter pieces remaining. Drizzle in the ice water, one tablespoon at a time, pulsing until the dough just comes together. Form into a disk, wrap in plastic, and chill for at least 30 minutes.

Prepare the Frangipane: While the dough chills, cream the softened butter and powdered sugar until light and fluffy. Beat in the egg yolk and almond extract. Gradually mix in the almond flour until fully combined. Set aside.

Prepare the Cherries: In a medium bowl, gently toss the pitted and halved cherries with granulated sugar, cornstarch, and lemon juice until evenly coated. Set aside to create a liquid.

Assemble the Galette: Preheat oven to 400°F (200°C). On a lightly floured surface, roll the chilled dough into a rough 14-inch circle. Transfer the dough to a parchment-lined baking sheet. Spread the almond frangipane evenly over the dough, leaving a 2-inch border.

Fill and Fold: Spoon the cherry mixture over the frangipane. Gently fold the dough edges up and over the filling, overlapping them slightly to create a rustic crust that partially exposes the filling.

Finish and Bake: Brush the folded crust generously with the egg wash. Sprinkle the crust with coarse sugar and the slivered almonds. Bake for 35-40 minutes, or until the crust is deep golden brown and the filling is bubbly. Let cool slightly before slicing and serving.