Tell her that she can have cheese when she eats the other things you have set out for her. Set a small bowl of cottage cheese to the side...out of reach...and then offer her the other food. She can't have the cheese until she eats the other food. Do not give in. Do not succumb to her tantrums. A 15 minute tantrum is nothing. And crying and screaming might get on your nerves but it is not damaging them in any way.
In our house the tantrums were not even allowed. I told the girls what their options were and they decided without incident or they spent their tantrum in their crib. I didn't care if they screamed for an hour. I would occassionally go in the room and tell them when they stopped screaming they could come out, but tantrums were not allowed. Screaming to get your way was not going to work.
When they calmed down I very sweetly told them that I was the mommy and I made the decisions. I would hug them and love on them and tell them that they are not allowed to scream to get their way.
I started this with the first tantrum....with my oldest that was probably about 10 months old and the second child it was maybe a little younger. THey understand most of what you are talking about and they definitely learn that they won't get their way by screaming.
My five year old has started stomping her feet on occassion. And she is reprimanded very quickly when I see her do it. She is not allowed to throw tantrums in any form. I tell her that I always listen to her and what she wants, but I make the decision after I listen and that's that.
If you don't nip it in the bud now, you will have a screaming, in your face, door slamming teenager on your hand with no respect for you or your authority as a parent.
Don't worry she won't be scarred if you let her cry it out in her room for whatever time it takes.
You have to convince her that you are the boss and she is not. It will work out fine and you will be happy you got more stubborn than she is.