To prepare cookie base: Preheat oven to 350 degrees and grease the bottom and sides of a 9- by 13-inch baking pan with butter.
In a medium bowl, whisk together the flour, baking powder and 1/2 teaspoon salt, and reserve.
In a bowl of an electric mixer on medium-high speed, beat 3/4 cup butter and 1 cup granulated sugar until light and fluffy, about 2 minutes. Add egg and almond extract and beat until combined. Reduce speed to low, add flour mixture and mix until dough becomes clumpy. Using your hands, pat the dough evenly into the prepared pan, smoothing it as much as possible to a uniform thickness. Bake until dough is just set and the edges begin to brown, about 10 to 12 minutes. Remove from oven and transfer baking pan to a wire rack.
In a small bowl, lightly beat the egg white, then brush the cookie base with the egg white.
To prepare topping: In a medium bowl, combine the apricots, walnuts, almonds, 1/3 cup granulated sugar, lemon zest, orange zest and 1/4 teaspoon salt, and stir to combine. Add 4 tablespoons butter and use your fingers to mix ingredients together. Spread topping mixture evenly over the par-baked cookie base. Lightly press down on the topping mixture to help it adhere to the cookie base. Return baking pan to oven and bake until nuts are golden brown and fragrant, about 15 to 20 minutes. Remove from oven and transfer baking pan to a wire rack.