Position rack in the bottom third of oven; heat oven to 350 degrees. Lightly butter a 8" square baking pan and line the bottom of the pan with parchment paper.
In a food processor, combine the flour, sugar, and salt. Add the butter and pulse until the mixture resembles moist pepples. Scrape the dough into the prepared pan and press to form an even layer. Bake until the crust is lightly golden, about 30 minutes, then transfer to a rack. Leave oven at 350 degrees.
While 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 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.