It's really great that you have such a good relationship with your therapist and that you want to be friends. This kind of feeling towards a therapist is often something called positive transference and many therapists see it as a good sign that the therapy is going well.
Good therapists also know that they can't ethically be friends, but it's very ok to want that. It's ok to share it with her too. You are not alone in this feeling at all, and it's quite common in therapy.
On the general topic of friendships with therapists, as others have mentioned, it's a big ethical no no for a therapist to have a friendship with a client. There are some really good reasons why that are not always immediately obvious.
I had a therapist that I was friends with. (Actually, my family was friends with her too, so it was doubly weird.) I had just started therapy, and I didn't really have a clue what was ok and not ok in therapy, and I didn't know there would be a problem with it. The therapy was good, but when we moved into being friends, the entire relationship self destructed in a pretty profound way.
There is something strange about having dinner at a restaurant and your therapist sits down and says hi wants to talk about her crazy conversation she just had with your uncle/her neighbor. I freaked. I didn't think anything like that would make me feel so bad. But it was like the person who knew all my trauma just showed up in public with all her own problems and needs in friendships and I freaked out. I didn't think I would. She didn't say anything about my trauma, but it just screwed up everything. (Yeah, she eventually lost her license too...)
Part of why therapy works is that it is a one way relationship. We are able to come to the room with all our needs, and we don't have to give back to the therapist in the same way we do in healthy friendships. We are able to project onto our therapist all kinds of stuff to work through it. The therapist can self disclose, but if they are doing it right, they seek to do it in a way to help us connect with them and feel connected enough to work through our own stuff, rather than to work through their own stuff and get their needs met.
In my life offline, I have two friends who are therapists. They have never been my therapist and we keep very strong boundaries around them even acting "therapist-y." Occasionally one of us will say, "ah, I think we need to take the therapist hat off." They are very wonderful people. I bet that some of their clients want to be friends with them. As friends... well, I couldn't ever do effective therapy with them. Ever. I just can't seem them in that way, because I know them like friends. In friendships, the relationship is two ways. There isn't the safety of privacy, and the normal non-therapist needs of my friends changes the way we relate than if we were in a therapist-client relationship. I see them on their own worst days. I can't process past trauma very well with someone who I know just broke up with their boyfriend or spent the whole night crying over their mother-in-law being insulting of their cooking again.
In therapy, we are in a place where we are listened to, heard, respected, safe, validated, understood, where we can laugh and connect in safety and where therapists often can relate to things we are going through (much more than many therapy clients realize)... who wouldn't want to be friends with someone who is like that? This pull to want to be friends with someone like that can be especially strong if we didn't get these normal natural healthy needs met as kids or we were otherwise wounded or traumatized in relationships.
I think it's a good idea to talk to her about how you do know the boundaries of therapy, you respect them, and that you still wish you could be friends. The fact that you wish for friendship where you can connect with someone in the same ways the two of you jive together so well - this is a legitimate and healthy thing to want. Maybe she can help you find that with other people in your life where you can be friends and get those needs and desires met.