Frame it another way. Completely forget your current situation and think of it like this. 

Lets say your girlfriend has a male best friend. For some reason, you do not trust her male best friend and you are suspicious something is going on. You love her, but he is really sketching you out. 

Now lets say your girlfriend went to this male best friends house, alone, and cooked him dinner. You don't know what she cooked. You just know she cooked. You have minimal information about how the rest of the evening went by. She says they just hung out. 

How would that make you feel? Your very first feeling. Without thinking about it. What would your reaction be? Would you be suspicious? Hurt? Upset or angry? 

Relationships are weird. There are generally "taboo" things you just don't do when you are in a relationship. Those things change depending on the comfort and boundaries both partners put forth. 

You are in a tricky spot. This person has been your friend for a long time. Its perfectly fine to have female friends. Where you become inappropriate is disrespecting your relationship with your girlfriend. She has voiced concerns /accusations of cheating in regards to this particular female friend. She has a reason, justified or not I cant say, that she does not feel comfortable with this particular friend of yours. Cooking dinner 1 on 1 with a woman your girlfriend is uncomfortable with is disrespectful to your girlfriend. I honestly don't know many women who would feel okay about this particular situation. As dinner 1 on 1 is usually an intimate thing typically viewed as a date. To some people, a situation like this could loosely be viewed as cheating aka you went on a date with another woman. Just as if she went on a date with another man. 

The situation changes if its you, female friend, and a third person of any gender. Its changes the stereotypical "intimate" dinner plans into a group hang out. It takes away the "date" aspect of cooking 1 on 1 for your friend. This is the option that is most respectful to your girlfriend/relationship. 

Its perfectly fine to be friends with this person. A relationship should never dictate your friendships. However set and setting play a role and certain settings lead to disrespectful situations that will cause problems in your relationship. 

You should absolutely be able to cook with /hangout with your friends regardless of gender. But, while in a relationship, you also need to be mindful of your actions and the situations they cause and how those actions will affect your girlfriend. Just as she should be mindful of her actions and how they affect you. There should never be double standards. And neither of you should put yourself in 1 on 1 settings that are making the other uncomfortable or upset. This is where compromise comes in. She doesn't want you solo with this female friend in date like settings. You don't want to lose your friend (who would?) So you make the effort to avoid date like situations while 1 on 1 or hang out in groups or include your girlfriend in hanging out. 

Lastly, if your girlfriend and this friend become friends or get to at least know each other, its possible she won't feel so insecure or suspicious about this friend.