For chickens, pasture is a supplement to their other feed, not a total replacement. Having legumes along with the grass helps, but they need a lot of insects, worms and larvae to replace a formulated feed. If you live in a tropical climate with a lot of insects all year or you have other animals around that they can steal food from, you might be able to get away with not feeding them. For the majority of people, you need to offer a complete feed.
The eggs do have a better fatty acid ratio, better cholesterol numbers and more vitamins in them. Mother Earth News had an article on this and there have been some other studies I've read about online. There's also a lot of information online these days about the nutritional differences in meat from pasture raised animals like beef. If you do a search on the forum and on google, you'll get a lot of information on some of these topics.
The healthier eggs come just from letting them free range. You don't need to restrict their regular feed, to force them to eat green feed. Chickens confined to a run can also benefit from having green feed offered to them.
If your birds were totally free-ranged on good grass then all they would need is a little (very little) extra feed.
I free range all my young cockerels from age 12 weeks until they don't get along anymore. I give them just enough whole corn to keep them coming to the barn so that I can catch them later. They get what they can clean up in about 1 or 2 minutes. That is it. They have to look and hunt for all the rest of their feed.