Eggs are used mostly in holiday sweet breads.
Its effect on bread,on top of giving it a richer taste, is helping the growth of the dough and giving the crumb a fluffy and soft texture.
When I bake the panettone, or Italian Christmas bread, according to the formula presented on this site, I usually give it a longer fermentation time and give the final piece a longer proofing time so that it grows much more than a piece of regular everyday bread. The egg gives more support to the dough and allows it go much higher without collapsing.
Also, during cooking the egg coagulates. This results in a tender crumb which is not at all sticky. When cutting slices of the bread will notice that the knife is clean.
The presence of the egg causes the crust take a golden brown appetizing sight.
When the eggs do not give color: The eggs we usually purchase in grocery stores are quite pale and give very little color to the dough. A festive bread should have a nice orange-yellow colour in its crumb, but it becomes pale and faded when using commercial eggs. Eggs from hens fed the traditional way, have a richer color and taste better.
It is common to use turmeric (Curcuma longa) to intensify the color of the dough in which commercial eggs have been used. Turmeric comes from a root (rhizome) similar to ginger and like it, has healing properties. It originates in southern India, where it is widely used. The addition of very small amounts of turmeric does not affect the quality or diminish the nutritional and health value of bread. The bread actually becomes a bit healthier.
The amount to be used in a homemade recipe is very small. Say: in six cups or 1 kg flour, the eighth of a teaspoon would be enough.
It is difficult to measure very small quantities in home baking. So the turmeric can be diluted in flour. Put a tablespoonful turmeric and seven tablespoons flour in a small bowl and mix them thoroughly. A teaspoon of the mixture–which is easy to measure–equals 1/8 teaspoon pure turmeric.