Preheat oven to 350°F. For easy cleanup, line an 8-inch baking pan with foil. Spray the foil with nonstick cooking spray and set it aside.
In a large bowl, whisk together both flours, the oats, brown sugar, salt and nutmeg until well combined with no big clumps of brown sugar remaining. Stir in the melted butter, applesauce and vanilla until blended. Gently stir in the apricots.
Spoon the batter into the prepared baking pan, pressing it flat with the back of a spoon or spatula.
Bake for about 25 to 35 minutes, until golden.
Remove from the oven and let the pan cool on a wire rack for 5 minutes, then lift the slab of bars out with the foil out and let the bars cool completely on the wire rack.
Transfer the foil and bars to a cutting board and use a large knife to cut into 16 even squares.