On each side of the frame, fasten 1 additional piece of wood, using the protruding pieces as support. 
 
My technique for preventing the screws from showing: 
 - First, always screw from the rear of the frame, if possible 
 - First, pre-drill the area where you're going to screw with the 3 mm drill bit, to prevent the wood from splitting when you screw. 
 - Then, once the hole is made, take the 9 mm drill bit and drill over it, but only a few millimetres deep, so that the screw head can sink in slightly.