The microwave sits in the island cabinet. The door in front is shaker style. The inset door bead was applied to cabinet. The customer wants this door to lift up and slide in. Is the solution pocket door hardware?
From contributor J:
Similar applications use a pocket door hinge on horizontal.
When the knob is pulled, the lower edge lifts out and up, pushing the upper pins straight back in the groove, and the center pins go straight up until the door is parallel to the counter/cabinet bottom. Then it can be slid back in its horizontal position to stops to keep the knob from hitting. Tighter opening to door tolerances are needed - not loose tolerances, or the door will twist and may bind. A little wax once complete makes it surprisingly smooth.
Mock it up to see how easy it is. The routes run parallel to the vertical edge and to the upper horizontal edge, and meet at 90 degrees. We like brass pins at 1/8" diameter for 3/4" work, 1/4" for 1" doors. At times we mount the pins in strips on back edge of the doors to get the geometry a little better.
This is surprisingly easy to make and operate, and requires nothing more than know how, and no hardware. It also showcases skills over buying a solution, and as such helps set your work apart from the others.