Enriched breads

  If you replace the egg weight with water, you'll be reducing protein in the bread, i.e. gluten. 

The quality of gluten in dough is very important:

• If gluten is too weak it can’t stretch in thin films around the air bubbles produced during fermentation. The gas bubbles would then swell and burst, causing the loaf to lack volume.
• If gluten is too strong then it won’t stretch so the gas bubbles can’t expand causing a very dense loaf.

Cite

  I don't know what your whole recipe is, but it sounds like you may already know what you need (knead?) to do.  Good luck and let us know how it turns out.

Post your formula and method and we can help you.  challah usually contains much more eggs and is typically a very light bread so I doubt the eggs or reduction in eggs will make a difference.  Type of flour, hydration level and mixing methods will all effect the final outcome.

... just FWIW, my regular enriched dough has 2 eggs for 500g flour (plus milk - varying amounts depending on the dough's final use, but I weigh the eggs and milk together - typically 300-350g of egg+milk). then 80g sugar and 80g unsalted butter. (7g dried yeast + 7g salt) I use that to make a variety of buns, rolls, pain viennois etc.

This sort of thing:

I've never made an actual loaf out of it (other than pain viennois) though.

-Gordon

One thing I notice is that there is almost no rising time for the recipe. You mix it and immediately place it into the refrigerator. Depending on how warm the dough was and how cold the refrigerator is, it might not rise much. Plus, you've already divided the dough, so it will cool off faster.

I would suggest letting the entire mass of dough ferment for a while after you mix it, perhaps letting it double, then punching it down and refrigerating undivided.

Also, scalding the milk before using it will improve the crumb a bit.

If you want a lighter, fluffier bread, you are going to have to spend a little bit more time on kneading. The butter and the egg yolks, although not representing huge quantities in the recipe, are going to work against the rising of the bread, as fats tend to weigh down on gluten development (though they bring a necessary moisture and creaminess to the crumb party). When we make brioche in a traditional bakery, the kneading is done in two steps : first, without the butter, then with it. Total kneading time is upwards of 30 minutes (in a big, powerful machine). If you are mixing by hand, 7-9 minutes will not suffice to get you the airiness you're after. Granted, you're not putting in as much butter as a brioche, but 7-9 minutes for something super light and airy seems insufficient, especially with the formula and process you have there (yes, natural sourdoughs can be light and airy with much less kneading, but enriched breads are a different animal).

If you are mixing by hand, cancel all of your appointments after you begin kneading : you're gonna be at it a while (I exaggerate, but seriously). And I would recommend employing a kneading method that actively incorporates air into the dough. Turning, folding and pushing doesn't seem to incorporate a ton of air, while a slap and fold method will bring a decent amount into the dough. I would spend a good 20 - 25 minutes beating up the dough before calling sending it off to ferment on its own. That should give you very solid gluten development to trap all that gas, giving you a light and airy crust. If you are using a machine, let that baby run for 15-20 minutes before letting it ferment.

Also, before sticking it in the fridge, its a good idea to let it bulk ferment a little while at room temperature (20-30 minutes, at least), just to make sure things are on their way.