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It's not plastic. It's called thermofoil which is basically aluminum foil like you wrap food with. The process starts with a door blank made from MDF (medium density fiberboard) and a design is routed into the door face. The design usually is meant to mimic a raised panel to some degree but it is a 1 piece door blank or slab some people call it.
The door blank is heated to a high degree, glue sprayed over it, and the foil is laid over the back side and the excess is trimmed. Next the face goes on and folds over the sides and conforms to the shape of the routed out design. That's the quick explanation.
Thermofoil from not so long ago is known for peeling in time. Especially just over the cooktop or stove. Today's generation of thermofoil doors should last a lifetime. They are durable and heavy which is why they are first choice for new apartment buildings. Cause the tenants can beat the hell outta them and can't hurt them.
Bottom line is they are not repairable. I know that's not what you want to hear and you're going to try anyway but that's just the way us humans are. Have to see for ourselves.
Replacing them would be your option, good luck finding a place to buy them. Duplicating yours will be near impossible as cabinet door manufacturers change door styles every couple years just like automobile manufacturers do. And the cabinet manufacturer is the one who influences the cabinet door manufacturer to make that change.