In a large bowl, whisk together the flour, powdered sugar, and salt. Cut in the cold butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs. Gradually add the ice water, one tablespoon at a time, mixing until the dough just comes together.
Form the dough into a disc, wrap it in plastic wrap, and refrigerate for at least 30 minutes. This chilling time is crucial for preventing the dough from shrinking during baking.
On a lightly floured surface, roll out the dough into a 12-inch circle. Carefully transfer the dough to a 9-inch tart pan with a removable bottom. Press the dough into the bottom and up the sides of the pan. Trim any excess dough and crimp the edges.
Line the crust with parchment paper and fill it with pie weights or dried beans. Bake in a preheated oven at 375°F (190°C) for 15 minutes. Remove the parchment paper and weights, and bake for another 5-7 minutes, or until the crust is lightly golden. Let the crust cool completely.
In a large bowl, cream together the softened butter and granulated sugar until light and fluffy. Beat in the eggs one at a time, then stir in the almond extract and almond flour until well combined.
Pour the frangipane filling into the cooled tart crust, spreading it evenly. Arrange the quartered figs over the frangipane in a decorative pattern. Drizzle the honey over the figs.
Bake in the preheated oven at 350°F (175°C) for 30-35 minutes, or until the frangipane is golden brown and set. The figs should be soft and slightly caramelized.
Let the tart cool completely in the pan before removing it. Brush the top with milk and sprinkle with sliced almonds. Dust with powdered sugar, if desired.