Your cooking made me happy.
Is this correct? Can "your cooking" be used as a noun? The more I read it, the weirder it sounded to me.
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Yes, it is grammatical.
Interestingly, "your cooking" can mean either "the food you made" or "the fact that you cooked."
So, for instance:
"Everything you made for the care package was delicious. The food brought me great comfort during the stress of exams. Your cooking made me happy."
"I know it was a big deal for you to take the time away from your job to do this dinner. It means a lot to me that you did all the cooking personally. Your cooking made me happy."
It does sound a bit weird to my (American) ear, but not for grammatical reasons. The "-ing" form of a verb can be used as a noun. This "noun" form is called a "gerund".
I can imagine the example sentence being used in a scenario like this:
Older mother: Looking back, what did you enjoy as a child?
Grown up child: Your cooking made me happy.
If the conversation were between a man and a woman he was courting, or between a husband and wife, I would not expect this sentiment to be expressed the same way. A statement in the present tense would make more sense, like:
Your cooking makes me happy.
The way to a man's heart is through his stomach.
or a woman might say
Someone needs to write a book On the Care and Feeding of Boyfriends.
I also know families where the father does much of the cooking. In those families, I can imagine a child someday telling her father, "Your cooking made me happy."
The -ing form of a verb may indeed be deployed as a noun (when this happens we call the -ing form a gerund), and play the same syntactic roles as any other noun. In those roles it may still take the same sorts of complement it takes as a verb, and it may take either adjectives or adverbs as modifiers.
Cooking rice is boring.
I hate your cooking.
I love my mother, but her atrocious cooking drove me from home.
I must thank Cedric for heroically cooking such an enormous meal on such short notice.
Your cooking made me happy
But the problem is the cooking in contemporary (American) English can refer either to
(a) the act of cooking
(b) the result of cooking
In the sentence you've made I would think it means (a), or to reword it:
That you cooked for me made me happy
That you always cook[ed] for me made me happy
If you want to express the other meaning, then I would say:
I was happy to eat the food you made
Eating the food you made made me happy
Cooking may be a noun referring to the art of preparing food: "The artful way you prepare food makes me happy."
Cooking may be a gerund. Then this is a case of possessive with a gerund. It is like: "I hate his singing in the shower" – I hate that he sings in the shower. In your case: "It makes me happy that you cook." I guess that both forms could be understood as either "I am happy that you cook and nobody else but you" or "I am happy that you stopped playing and started cooking."