Cut the base pieces for the doors from the ½” plywood and from the ¾” plywood. Secure the ½” plywood to the ¾” plywood with one long edge and both short edges flush (the ½” plywood will overhang one end by 3/8”) using glue and 1” brad nails.
Cut 2-1/2” wide strips from the ¼” plywood. Position the first strip at the upper inside corner of the door base setting it at a 45 degree angle. Secure using glue and ½” brad nails. If the strips are cut a little longer than the door, they can be cut flush to the base using a router and a straight flush-cutting router bit. Repeat for all of the door fronts noting that there will be three right doors, and three left doors.
Install the hinges according to the manufacturer’s installation instructions. The ¾” portion of the door base will fit inside the opening with a 1/8” gap around all sides as well as in between. The 3/8” portion with the herringbone pattern will sit on top of the sides and dividers, overlapping by 1/4”. After installing the hinges, make any necessary adjustments.