This tart showcases the fragrant apricots with the almond flavor in both the filling and crust.
To make the crust: Preheat oven to 350 degrees with rack in center. Brush a 8-by-11-inch tart pan with removable bottom with butter.
Place the almonds and 1 tablespoon sugar in the bowl of a food processor, and pulse until the almonds are finely ground. Add the butter, and process until combined. Pulse in the flour, the remaining 1 tablespoon plus 1/4 teaspoon sugar, and the almond extract until combined.
Transfer the crumbly dough to the prepared pan. Using your fingers, pat out dough evenly to make a thin crust along the bottom and up the sides of the pan. Chill the crust 30 minutes.
Bake crust until golden, 20 to 25 minutes. Transfer to a wire rack to cool.
To make the filling and assemble the tart: Preheat oven to 350 degrees. Spread the almond slices in a single layer in a rimmed baking sheet; toast in oven until golden, about 10 minutes. Shake the pan halfway through baking to make sure the nuts toast evenly. Set aside.
In a large bowl, combine 1/2 cup granulated sugar, salt, all-purpose flour, almond flour, and baking powder; whisk together. Whisk in the eggs, melted butter, and half-and-half until well combined. Pour the filling into the baked crust.
Arrange the apricot halves, cut sides up, on top of the filling, packing the fruit closely together. Sprinkle the tops of the apricots with the remaining 1 tablespoon granulated sugar. Sprinkle the toasted almonds over the entire tart.
Bake the tart until the filling is puffed and golden brown, 60 to 70 minutes, rotating the pan after 30 minutes for even browning. Transfer tart to a wire rack to cool. Unmold, and sprinkle with confectioners' sugar just before serving.