Painted Kitchen Cabinets
If the paint is peeling off the wood surface then most likely there was a problem with how the cabinet was prepared for the paint or with the paint itself.  Painting any surface located in a kitchen takes more effort and attention to details when prepping it for paint. As you already know grease from cooking that becomes airborne in water vapor will adhere to wall surfaces and accumulate over time. For proper paint adhesion it must be removed through scrubbing with a de-greaser-type cleaner. Most commonly used is the cleaner labeled or containing "TSP" (Tri-Sodium Phosphate). Added to warm water and applied with a sponge (or sponge mop) the cleaner easily cuts through greasy build-up. Rinse the applicator often and then re-wipe with fresh clean water to remove any residue.  Since the cabinets are already peeling it will be hard to hide the edge of the paint with a new coat. You can try sanding the edge to "feather" or angle the transition so when another coat is applied it may be less noticeable. But I would guess since the wood hadn't been prepared you will continue to have peeling issues. It may be wise to start anew by sanding all the door faces to remove new paint, clean with TSP and prime and finish coat.