The original author is in error adding baking soda (bicarbonate), and indicating it will help rise. For leavening, you require acid for the sodium bicarbonate to react with. The original base recipe has no significant acid ingredients; even the modified version you used has only a trivial amount of acid from the molasses in the brown sugar.
Therefore, the only effect of the baking soda is to increase the pH of the dough, which will encourage browning. However, as the cookies are full of sugar, encouraging browning is not generally a problem.
The creaming method creates bubbles in the solid phase butter, and encourages rising and a more cake-like result, especially when paired with an effective leavening agent.
Resting the dough also hydrates the flour, and allows the butter to re-solidify, both of which inhibit spread and contribute to a higher cookie.
For a chewier result, do not use the creaming method. Melt the butter instead, and then combine your ingredients.
You may also choose to reduce the baking soda to reduce the metallic taste it brings without anything to react with.
If the cookies are now too dense, you might try adding (starting with smaller amounts) 1/2 tsp to 1 tsp or so of baking powder which has its own acid to react with, to help leaven the cookies.
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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
Some keys to keeping cookies thick, beyond the chilled dough, are: use a starchier flour, use a fat with a high melting point, and use baking powder.
A starchier flour, like cake flour, will spread less than a flour that has higher protein, like all purpose. If you can't find cake flour, you could replace some of the flour in your recipe with a starch, like cornstarch or potato starch.
Shortening has a higher melting point than butter, replacing all or some of the butter in your recipe with shortening will allow the cookie to begin to set before the fat melts completely.
Finally, baking powder contains acids which help to keep the dough tighter. This helps it to "puff" more and spread less. If you can't use baking powder, you could add a bit of some other acidic ingredient to your dough (but cut the baking soda a bit as well or you might end up with cookies that are too cakey). A bit of a fruit vinegar, citrus juice, or cream of tartar would probably do the trick.