My immediate thoughts are to use the following ingredients:

Almond Flour, Sugar, Salt, Baking Soda, Applesauce, Vegetable Oil, Vanilla Extract, and chocolate Chips.

Almond flour is safe to eat raw (as opposed to regular flour which isn't), as are applesauce vs. eggs. This has two advantages- avoiding opportunities for food borne illnesses, and not cracking/mixing eggs.

Vegetable oil is easier than melting butter. It may lead to a greasy cookie, but it also won't stick to any parchment paper as badly.

Further, it is, in my opinion, ill advised to use measuring tools. Your hands can approximate cups, 1/2 cups and the like. Pouring applesauce and vegetable oil can be done by eye (carefully, but quickly). Mix by hand in a single bowl would have to be the fastest method.

As for baking, to avoid burning yourself it might be wise to preheat a cookie pan on top of an oven as opposed to inside it. The pans won't be cold and suck heat in when warming, thus (if I'm not crazy), leading to slightly faster baking time.

But, uh, this is all untested conjecture. Saw this category on Reddit and wanted to throw in my $0.02.

Few things,

I reworded the Butter ruling. I realize it was a bit confusing and made it seem like melting the butter was required. It is not required.

I would advise doing a lot of testing before the run. See roughly how much product you can hold in your hand. weigh how much you grab on average by hand.

Maybe do a rest bake with all of your ingredients. See that the recipe holds up and meet what would be considered an acceptable cookie. Feel free to post pictures in the forums to see if it would be accepted or not.