Eat gracefully and pay attention to fatty acids

Please read this interesting article:
http://mobile.nytimes.com/2014/03/26/opinion/bittman-butter-is-back.html?referrer

But how can a vegetarian take all the essential fats? Are cheese, yogurt, eggs, avocado an nuts enough?

There are 2 important steps vegetarians may take to improve EFA status. 1) Maximise the conversion of ALA alfa linoleic acid to EPA and DHA. 2) Provide a direct source of EPA and DHA.

If your need for EPA and DHA is increased (eg, pregnant and lactating women) or at greater risk for poor conversion (diabetes, neurological disorders, premature infants, the elderly), it may be prudent to ensure there is a direct source of EPA and DHA.

The ones you mentioned such as egg, avocado, nuts belong to Omega 6. Over consumption of Omega 6 (pro-inflammatory) can pose problem and Omega 3 (anti-inflammatory) is very important. The balance between the two is the key.

And so ALA (which can be converted to Omega 3) is found in green vegetables for instance:

purslane
Brussels sprouts,
kale
spinach (beware of oxalate)
leafy greens

The other type, eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA), is found in fatty cold water wild fish. The body partially converts ALA to EPA and DHA. For good health, do aim to get at least one rich source of long chain omega-3 fatty acids in diet every day. This could be done through a serving of cold water wild fish:

Atlantic salmon
Mackerel
Atlantic anchovies
Sardines (excellent source of calcium too)
Tuna (beware of mercury)

Or

Soy (fermented natto, fermented tempeh)
Wild rice
Kidney beans
Chia

sprinkling a handful of walnuts mixed into the morning oatmeal,
seeds in the salad (sunflower, pumpkin, hemp, flax)
cold pressed oil of these seeds
using grains (millet, amaranth, quinoa)
eat legumes
algae (avoid large intake of seaweeds as it may contain excessive amount of iodine)

Flaxseeds is an exceptional source of lignans, a potent anticarcinogen and one the richest known source of the essential Omega 3 fatty acid. The question to consider for vegetarians (and vegans) is if they convert ALA alpha-linolenic acid to EPA and DHA in sufficient quantity (they tend to have too much Omega 6).

If you decide to go for fish oil supplementation, it’s worth noting that although fish oil supplements (EPA/DHA) are safe for most people, they do have anti-clotting actions and could be dangerous for people with blood clotting disorders or those taking anti-clotting medication.