What kind of nuts are eco-friendly?

Building on Chris' answer: think about transportation.
For example, California produces a huge percentage of almonds and other nuts, and they are exported everywhere: I live in Italy and for example it's much easier and cheaper to buy walnuts from California than from any other country (Italy included).

Probably buying local nuts it's the primary factor. Given that, you can think about the rest.

Interestingly the cashew fruits are edible and some people do cook them. Permaculturalists also eat the young leaves cooked.

I don't think yield of nuts is the only issue. The other issues must include long-term soil issues and the like. While higher yields can in theory support larger populations, that's only one long-term variable.

In general, nuts are probably more sustainable as a class than beans because beans are typically grown as pioneer species on disturbed soil while nuts are grown as longish term trees. While beans fix nitrogen, the fact that they are pioneer species works against them in the long run.

The reason why permaculturalists disfavor pioneer species for long-term sustainability is that they grow in disturbed soil, which requires ploughing, and this leads to both depletion of nutrients and erosion of soil. This requires strategies for mitigating both pests and nutrient depletion. Trees are considered more sustainable because they represent an ecological niche closer to the apex of ecological succession and are thus can be grown in ways which require, long term, far fewer inputs. Of course this is only one factor and how the trees are grown is critically important too. For more on this, see Toby Hemenway's Gaia's Garden.

In short there are a very large number of variables. Some that I think you should consider are:

1. How useful are the rest of the parts of the tree? With cashews, the "apples" can be cooked and eaten (mildly astringent fruit resembling the Indonesian jambu air). With walnuts, the wood is useful. We don't use as much hazelwood as we used to (it's good for bows and for tally sticks). And so forth.

2. What is the impact long-term on the soil as it is typically grown? How much fertilizer goes in? How much in the way of pesticides? This is a real issue and it may differ so much from region to region that it makes a clear comparison difficult. In general organically grown nuts as a class will do better than non-organic.

3. What is the embodied energy of the labor? Does it require a lot of migrant labor which comes in via car and moves around (in small groups by car) around a large area every year? Or does it support more sendantary labor? Tropical nuts will do better than temperate nuts in this regard, but....

4. Are you growing them yourself? There is no more sustainable food than the food you grow and harvest yourself without chemical fertilizers or pesticides.

I question your basic premise that a plant's edible yield density necessarily equates to "eco-friendly." That may turn out to be the case, but it would have much more to do with the conventional farming and production practices used, offset by any possible other sustainable uses for these disposable byproducts.

Consider that most nuts come from trees that are huge — dozens of feet tall — and often live for hundreds of years. Barring other external factors (like clearing rainforests to create farmlands, or indiscriminate use of fertilizers or pesticides that may destroy eco systems), harvesting fruits from long-lived tress seems eminently more eco-friendly than a conventional "instant crops" like beans, peanuts, etc.

But to answer the crux of your question, cashew trees produce on average about 817 pounds per acre (916 kg/hectare) (source). Compare that to other commercially-grown nuts —

From Profitable Plants:

• Almonds — 1,500 to 3,000 shelled pounds per acre per year
• Chestnuts — 2,000 to 3,000 pounds per acre
• Filberts — 1,500 pounds of dried nuts per acre
• Pecans — 1,200 to 2,000 pounds of nuts per acre
• Walnuts — 6,000 pounds per acre