That isn't' a cabinet issue. That's a cooking method issue. Get a splatter screen, and turn your temperature down. There's no way that anything should be splattering that high.

Came here to say the same, wow that’s a high splatter! splatter!

Agree with both posts above.

Here's what I would do, based on experience of past paint fixes. I would suggest cleaning them several times with Krud Kutter, plus a final water wipe. Once throroughly dried out (give it a day), you can prime with BIN shellac based primer or Kilz oil based primer which should both handle any remaining oily residue/stain bleed-thru. Get a sample size paint pot color-matched to a cabinet door and repaint, then do a couple protective top coats over the paint. It won't need to be a fancy cabinet paint type since you are top coating it- that will become your final layer. Or for maximum protection and future ease of cleaning you may want to get some thin plexiglass cut to size and screwed over these side panels.

I hardly get splatters on my backsplash tile. You must be an aggressive cook.

For the time being, I'd carefully clean that side portion of the cabinet and then cut a sheet of acetate to size and carefully side it between the hood and the cabinet. I'm thinking it might stay just because it's a tight space.

Acetate is the clear material used for crafts or to protect items in binders.

https://www.michaels.com/product/clear-acetate-sheets-by-recollections-12-x-12-10656641

Is the exhaust fan working correctly? Is it recirculating or ducted to the outside?

My DH is not a careful cook and he would probably get splatters in those areas too! Having a hood that is wider than the cooktop helps to keep stuff away from cabinets.