Neutral non offensive alternative to "slanting eyes"

"Slant eye" would mean a person with slanting eyes, and you'd be describing the person, not their eyes.

It's offensive because it was used a derogatory term from the start, much in the same way that the N-word is offensive to Black people. On the surface the N-word is simply a word to describe a Black person, but because of its historical use it is extremely offensive.

The typical way to describe someone who has those characteristics is to simply use their country or region of origin. For example, 'That man is Asian' or 'That man is Vietnamese', etc.

Or, better yet, don't just generalize a person based on their physical appearance and actually ask the person where they come from. For example, an American with Chinese parents would display those characteristics but be an American.

You might consider, have Asian eyes

If, for some reason, you need to refer to the eye shape of a person with Asian heritage (or with similar eye shape), you can refer to the epicanthic fold. Less technical, but slightly riskier: almond eyes.

Current style is to avoid references to race, skin colour and such like in most situations. If you have to point out the one Chinese-American in room full of Swedish-Americans you might have to refer to the lady with the green shoes or something like that, and 'person' would be safer than 'lady'.

Terms such as "slant-eyed" and "mongoloid" are almost always very offensive.

If you're solely discussing shape, I've seen them referred to as almond-shaped eyes, but I have no idea if this might be considered offensive to people who aren't round-eyed.

I agree with @Spehro Pefhany that the safest term these days is "epicanthic fold". It is very unlikely to cause offense.

However, I would add that technically speaking, the epicanthic fold is not necessarily present in all Asian races. There are many who have the 'slanting' [sic] eye shape but who do not have the epicanthic fold. I agree that it is a more useful term than most others which can be construed as pejorative. But strictly speaking, an Asian person might not have the epicanthic fold and a Causasian person could have it, for example. Just something to note if you decide to use this term.

I wonder if the terms 'elongated eye shape' or 'oval eye shape' might be viable alternatives.