Why don't you cook mutton? Correct
Why don't you cook beef? Correct?
What do you mean by "grammatical"? How does a native speaker say those sentences?
Naeem, in what circumstances do you want to use those sentences?
For, example, if you want someone to cook chicken, then neither is correct. I know that's pretty obvious in this example, but it often isn't. Sometimes we get strange sentences that could be more or less correct, but we don't know what the writer thinks they mean. All we can say is that they are grammatical.
Yes, that is the right way to say it.
Why don't you cook mutton?
I mean I want to ask someone why she/he doesn't cook mutton (meat from the goat and sheep) whether she/he dislikes it.
Why don't you cook beef?
I mean I want to ask someone why she/he doesn't cook beef (meat from the cow) whether she/he dislikes it.
We are far more likely to ask, "Why don't you eat mutton/beef?"
Can I say: Do you eat mutton? Is it natural?