The Hebrew word for sabbath  comes from the word "shavat," which means "ceased."
We are to cease our work (occupational / gainful working) on certain days.  I give quite a few scriptures on this point in my study "Our Pleasure on the Sabbath?" .  There are no actual, clear statements in Scripture that say one cannot cook on any Sabbath (day of ceasing).  If you are a cook and your job is cooking then, no cooking FOR PAY.  But, what we are ceasing is gainful working;  because in the eternal rest, we will not work to earn a living.  Cooking your own food is not your occupation through which you earn a living or extra money for your wants.

What about:

Exod 16:23 (KJS) And he said unto them, This [is that] which {Yahweh} hath said, Tomorrow [is] the rest of the holy sabbath unto {Yahweh}: bake [that] which ye will bake [to day], and seethe that ye will seethe; and that which remaineth over lay up for you to be kept until the morning.
Notice it doesn't say bake ALL of it so you do not have to bake tomorrow.  It says bake what you will and boil (seethe) what you will, and what is left (what you have not cooked) keep for tomorrow.  They were being allowed to keep extra for tomorrow, whereas on other days of the week they were not to do that or it would spoil.
Exod 16:19 (KJS) And Moses said, Let no man leave of it till the morning.
20 Notwithstanding they hearkened not unto Moses; but some of them left of it until the morning, and it bred worms, and stank: and Moses was wroth with them.
So, they were not to gather it on the Sabbath, but get twice as much on the sixth day (kind of like, today; get enough on Friday so that it is not necessary to buy food on the Sabbath); but it really doesn't say that you can't cook it on the Sabbath.  I think that it's possible that Exodus 12:16 is confirming this:
Exod 12:16 (KJS) And in the first day [there shall be] an holy convocation, and in the seventh day there shall be an holy convocation to you; no manner of work shall be done in them, save [that] which every man must eat, that only may be done of you.
Verse 16 seems to be confirming that no one goes to work on a high day, but of course you have to eat and for that you may do the preparation of your food (without paid cooks or paid servers, of course).

Now I may be wrong, but this all seems to be the possible answer about cooking on the weekly Sabbath (day of ceasing), and high days (also days of ceasing).

I know that this position will upset some and make others down right angry with me, but I've been there before.  I do not think that Yahweh made rules for us, just to make us miserable; we must be careful how we evaluate all related scriptures on a subject. Adding rules just to be safe is not acceptable.  Scripture says NOT to add or take away from Torah (Yahweh’s guidance, direction, and teaching); see Deuteronomy 4:2.  Many assemblies have added to the Law (Torah), although very few take away; what is added is almost always restrictions, not positive things to do.  So with some, the body of  Law becomes larger and increasingly restrictive; but by man’s doing.  Does this sound historically familiar?

I try my best to understand, and may Yahweh guide us as we listen for his voice.