You will need two pieces of plywood, one the width of the framework you see, and another just slightly wider than the gap that will be formed by pulling the first piece of plywood out. The first piece of the bed will fit on top of the large piece of plywood, the second will fit on top of the smaller piece of plywood. That is going to be the minimum you need to have a double pullout bed there.
The larger piece of plywood will be screwed to the moving part of the frame, and the edge of the smaller piece will be screwed to the non-moving part of the frame, against the wall.
Meddle not in the affairs of dragons, for you are crunchy, and taste good with ketchup.