Position an oven rack in the bottom third of the oven and heat the oven to 350 degrees F. Lightly grease an 8-inch square baking pan and line the bottom with parchment. Combine the flour, sugar, and salt in a food processor. Add the butter pieces and pulse until the mixture resembles moist pebbles. Dump the dough into the prepared pan and press down to form an even layer. Bake until the crust is lightly golden, about 30 minutes. Transfer the pan to a rack and leave the oven set to 350 degrees F.

While the crust is baking, combine the brown sugar, egg, apricot jam, flour, baking powder, salt, and almond extract in a medium bowl. Whisk until blended. Stir in the apricots and coconut. When the crust is baked, scrape the topping into the pan and spread evenly. Continue baking until the topping is browned around the edges and a pick inserted in the center comes out a bit sticky, about 20 minutes. Set the pan on a rack to cool completely before inverting onto a cutting board. Using a warm, thin-bladed knife, trim off the edges and cut into bars.

Store at room temperature or freeze in an airtight container, separating the layers with waxed paper.