Preheat oven to 350°F. Place a rack in the center.
Make the crust: In a food processor, pulse 1 1/2 cups flour, 1/3 cup sliced almonds, 1/4 cup granulated sugar, and a pinch of salt until the almonds are broken down. Add the cold cubed butter and pulse until the mixture looks like coarse sand. Press it firmly into the tart pan — bottom and up the sides. Freeze for 15 minutes.
Blind bake the crust: Prick the bottom with a fork. Bake for 15–18 minutes, until the edges are just starting to turn golden. Let cool on a rack. Leave the oven on.
Brown the butter: Melt 1/2 cup butter in a small skillet over medium heat. It will foam, then quiet down, then turn the color of hazelnuts. Swirl occasionally. The moment it smells nutty and looks amber, pour into a heatproof bowl to stop the cooking.
Make the frangipane: In a medium bowl, whisk together 2/3 cup sugar and 2 eggs until smooth. Add 1 cup almond flour, 1 tsp vanilla, and 1/2 tsp almond extract. Stir in the warm browned butter.
Prep the peaches: Slice the peaches into 1/4-inch wedges. If very juicy, pat gently with a paper towel.
Assemble the tart: Spread the frangipane evenly over the cooled crust. Arrange peach slices on top in overlapping circles starting from the outside. Sprinkle with 1 tablespoon turbinado sugar.
Bake: Bake for 40–45 minutes, until the frangipane is puffed, golden, and set in the center. A toothpick inserted near the middle should come out clean. Let the tart cool completely in the pan on a wire rack — at least an hour.