Preheat the oven to 350 degrees F. Lightly coat the bottom of an 8×8-inch baking dish with nonstick cooking spray or line it with parchment paper.
Place the dried apricots in a small saucepan and add enough water to cover them. Bring to a boil, then reduce the heat slightly and simmer for about 10 minutes, until softened. Drain well, let cool, and chop.
In a medium bowl, use an electric hand mixer or food processor to combine the butter, granulated sugar, and 1 cup of the flour until coarse crumbs form. Press the mixture evenly into the bottom of the prepared baking dish. Bake for 15 to 18 minutes, until lightly golden around the edges.
While the crust bakes, use an electric hand mixer to beat the eggs, brown sugar, and vanilla until smooth.
In a small bowl, whisk together the remaining ⅓ cup flour, baking powder, and salt. Stir the dry ingredients into the egg mixture just until incorporated.
Fold in the chopped apricots, walnuts, and flaked coconut. Spread the mixture evenly over the hot crust as soon as it comes out of the oven.
Return the pan to the oven and bake for about 30 minutes, or until the top is puffed and golden and a toothpick inserted in the center comes out clean.
Cool completely, then cut into small bars. Remove from the pan and either roll in or sprinkle with powdered sugar to coat.