Preheat oven to 350 degrees. Coat the bottom of an 8" x 8" baking dish with non-stick cooking spray.
Place apricots in a small saucepan and add enough water to the pan to cover the apricots. Bring to a boil then reduce heat just a bit and keep at a low boil for about 10 minutes. Drain them and set aside to cool and then chop them.
Use an electric hand mixer or food processor to combine butter, granulated sugar and 1 cup of flour until it resembles coarse crumbs. Press the mixture into the bottom of the prepared baking dish. Bake for 15 to 18 minutes, till lightly golden brown around the edges.
Use an electric hand mixer to beat eggs, vanilla, and brown sugar till smooth. In a separate small bowl, combine remaining 1/3 cup flour with baking powder and salt. Use a wooden spoon to blend this mixture into the egg mixture. Stir in nuts, coconut, and apricots. Spread the mixture over the baked layer as soon as it comes out of the oven.
Return the pan to the oven and bake for an additional 30 minutes or until puffed and golden brown and toothpick inserted in center comes out clean.
Cool completely and cut into small bars. Remove bars from pan and roll them in powdered sugar.