In a large bowl, mix the flour, sugar, and salt together until evenly combined. Add the cold, cubed butter and use a pastry cutter to work it into the mixture until it resembles coarse crumbs.

Add ice water one tablespoon at a time until the dough just comes together in a shaggy mass. Form it into a disk, wrap it in plastic, and chill in the refrigerator for about 30 minutes.

In a separate bowl, beat the softened butter and granulated sugar until smooth and creamy. Add the eggs one at a time, mixing well after each addition. Stir in the almond flour, all-purpose flour, vanilla extract, and almond extract until thick and smooth.

Once the dough has chilled, preheat your oven to 375°F (190°C). Lightly flour your work surface and roll the chilled dough into a circle about 10–12 inches in diameter.

Transfer the dough onto a baking sheet lined with parchment paper.

Spread the frangipane filling over the center of the dough, leaving a 2-inch border around the edges. Fold the edges of the dough over the filling, creating pleats as you go around.

In a small bowl, whisk the egg with the milk for an egg wash. Brush this mixture over the crust. Sprinkle sliced almonds and sugar over the crust.

Bake in the preheated oven for 35–40 minutes, or until the galette is golden brown and the frangipane filling is puffed up.

Allow it to cool slightly before serving.