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1
To make the crust: Preheat oven to 350 degrees.with rack in center. Brush and 8-by-11-inch tart pan with removable bottom with butter.
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2
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.
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3
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.
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4
Bake crust until golden, 20 to 25 minutes. Transfer to a wire rack to cool.
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5
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.
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6
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.
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7
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.
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8
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.
