Egg yolks and egg whites have very different roles in baking. You can almost never replace them. And in the cases in which you can, you will end up with a different texture.
In this case, it won't be sure disaster to replace, but I would be very reluctant to do it. Flourless recipes are finicky. Flour holds stuff together. Nut flours don't hold anything together, they need a binding agent. Egg whites are a good binding agent. Egg yolks are not only not good as a binding agent (except in certain circumstances, for example in custards - but you don't have this here), they are even a lubricant and as such will interfere with the binding done by the egg whites.
If you substitute here, you will not only change the taste, you risk your cookies crumbling apart in your hands.
In general, don't replace stuff in baking recipes. They are hard to design right. Even if you have some issues (e.g. allergies), it is easier to find an existing recipe without the offending ingredient than to try to tweak an existing one to work with different ingredients. Eggs are especially hard to substitute. Such a reason like "not wanting to waste" loses its sense if what you end up with is bad cookies, which are a much worse waste of products. And besides, you can do many more interesting things with a yolk than just scrambled eggs. In fact, I am frequently throwing out whites because I have found no use for them after doing something with the yolks.
If you still want to go ahead and try if the cookies work for you, try using 3 whole eggs. 2 eggs are nowhere near enough to substitute for 4 egg whites. Not only are the yolks smaller than the whites volume-wise, they will also not deliver you the binding proteins you need so badly.
The texture of cookies is a complex interaction of many factors, including the size of the cookie, the temperature of the oven, the amount of leavening, the way the fat is treated, and so on.
The flat, thin cookie with the cracks on top that you describe and desire is achieved by slightly under-baking the cookies, allowing them to rise, and then taking them out of the oven. They then cool and deflate, causing the "cracks" and thinness.
The most likely culprit is that you are over-baking your cookies, and they are setting in the oven while fully domed up. They would be fine cookies, but more of a crispy style.
A secondary, and related, contributing factor would be having the oven temperature too high. If you don't already have one, an oven thermometer is an inexpensive and helpful investment to make sure you are baking at the desired temperature, as many ovens are off by a fair margin.
Note also that these are quite large cookies, baked with a "large" scoop, only six per tray. That helps achieve the fallen state. Don't make your cookies too small, if you are looking for this texture.
To answer your sub-questions:
- Creaming cannot apply to a melted butter cookie, as it requires a solid but plastic fat to incorporate air into the cookie
- The egg contributes structure (from protein), tenderness, and usually a fairly considerable proportion of the overall water in the cookie dough
- In this recipe, the baking soda is primarily present to promote browning, as there is very little acid for it to react with other than from the molasses in the brown sugar (it will leaven a little). Since these are "fallen" cookies, it is harder to get good browning and therefore flavor development. The soda is to promote that.
Best Answer
Adding a whole egg would help, but you might start with adding just one egg yolk. The additional fat will address the dryness which leads to the crumbling.
You may also try other fats like butter, cream cheese, or vegetable oil.
If even one egg yolk is 'too much' for your desired outcome, you might try a small amount of half&half.