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Taximes
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The first would be to take some time off before grad school and really dig into the subject you want to switch to. Find someone on campus who works in the area you're interested in and do research with them, even if it means starting with grunt work, and audit classes while you do that. Build up your background in the area, develop recommendations, and then start applying around in your new area. I've never seen anyone specifically do EE to phsyics, but I do know other people who have switched related fields.
The second, and more specific to you and from my undergrad background, is go to grad school for EE and do physics oriented work. Cutting edge EE, depending on the concentration, essentially boils down to applied physics and since you don't have a background in theoretical physics I imagine that's what interests you. There really is a lot of extremely exciting work going on pushing technology in new directions, and I honestly think anything physics related that interests you there's a place where you can apply your EE background to it. Talk to people in your department and really hash out what you like about physics and how you can do that coming from an EE background, I think you'll be surprised what's out there.
Why? At what level? Masters/PhD?
It's not uncommon, but not all that common either. You most likely will have to take undergrad classes to get base skills that physicists expect at grad level. I wouldn't be surprised if you're told that you need to pay your own way for a Masters to prove yourself, before they fund you for a PhD.
Grad school in the sciences is one thing: research. You research. You publish said research. That's it. If the idea of working on a problem, that only you in the entire world might find interesting, for months on end, then grad school is a great place. I love it. I love the freedom, and I love what I do. I love to learn. However, you need a reason to be there. Going to grad school "just because" or you don't like the job market right now are not good reasons.
I would recommend thinking about why you want to be in grad school, doing Physics, in the first place. If you get rational answers that you feel passionate about, then the process of getting those other hurdles out the way will be a lot easier for you. If you don't have those answers, then don't bother. You'll just drop out when the going inevitably gets tough.
To get into any physics program you'll have to take the physics gre subject test (can I assume you're applying to American Universities?). You need to either take all those physics classes (including quantum physics!) or teach yourself that shit.
It of course helps to have those relevant courses under your belt already. I have degrees in math, so going from that to statistics wasn't that much of a reach for me, but mostly because I had a lot of experience in statistics. Definitely talk to your advisor and he'll be able to give you a better idea of what classes you'll need to have under your belt.
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I'm in grad school for biomedical engineering, which is kinda a montage of a bunch of different disciplines, so we get people from all over entering in the program.
I'd say 80% were biomedical engineering undergrads, but the rest are like physics (these dudes usually do medical imaging research), EE, CS, even the occasional biology major.
The biology majors are probably most analogous to your situation. They come in with a vague connection to biomedical engineering, but they're really at a big disadvantage with the technical skills even though we require them to have a certain number of engineering-type classes from undergrad in addition to normal bio stuff (just like I'm guessing you would be missing a lot of the theoretical physics stuff). Usually they are able to stay somewhat on track with our grad program assuming they go into one of the labs that is more bio-centric, but they always end up taking a bunch of undergrad engineering classes to shore up the holes in knowledge.
I believe my program funds this though (assuming they're Ph.D. students... the terminal Masters students get to pay their own way). As I think about it, I think overall the students coming from other disciplines either take just as long or maybe like a year longer to get out of the program, but otherwise it's not a huge deal at my school.
I found out pretty much the same things people are saying in this thread... you do need to know a decent amount of physics to take the physics GRE and be considered by a grad school. I'd be interested to hear what your counselor has to say on your options, though.
I would suggest taking more time as an undergrad to explore the physics major. Unless finances are a concern, I think there's no reason to rush finishing your undergrad. You should look into how much longer you would need to stay, and if your school offers a physics minor (I don't know if this is offered anywhere, but if the major takes too long it's worth checking into) You might find physics as a major to be either exactly what you wanted or not quite, but it is definitely better to find out now. With a better physics background in your undergrad you will be more prepared to go on as a graduate student in the subject, as well.
This is probably way too early to ask (I imagine you haven't decided what specifically you would want to explore in exhaustive detail) but what subfield of physics would you be interested in researching? If I recall correctly, biophysics phd programs are a little easier to get into, I think they are more accepting of people coming from different undegrad backgrounds...you should look into this a little further.