On będzie gotował / On będzie gotować
The gender is the same, the meaning seems to be the same - but which version should I use?
On a side note, if you take a look at old Church Slavonic, there are two distinct future tenses, one being formed with the past tense stem and the other one with the infinitve. In Old Polish the grammatical distincion was gradually lost, but both forms remained in use until this day.
The option with the past stem is considered more elegant, and it also contains more information than the infinitive.
When talking to two people, a man and a woman:
Będziesz gotować - You will be cooking. (no linguistic specification who the speaker is referring to)
Będziesz gotował - You (male person) will be cooking. (The speaker is adressing the man only)
This is why I prefer to use the first option and also recommend it to others.
Also, using the second option with auxiliary verbs, doesn't sound quite 'right' to me, but maybe someone disagrees.
For example: On będzie musiał gotować
I would never say: On będzie musieć gotować
Whichever you prefer. Both versions mean exactly the same. Version with past tense stem is more „limiting” because it must fit the gender.
Forms with infinitive can be also more common than it's counterpart:
Example with „it will be raining” (but also snowing, because it literally means: it will be falling) - Będzie padać: Google shows 130k - Będzie padało: 23k