When I strum hard, the low E string seems to slap on the frets (when played open). According to the info I was able to find, my action is within spec, so I'm guessing my problem lies at the nut end? Could a too-deep cut in the nut cause this?

Action being within spec does not mean strings wont ever buzz. You should first ensure your technique is correct, and you are not lifting the strings when plucking. If you hear strings slapping, that means you are probably lifting them quite a bit as you strum.

You want to strike the string as parallel to the fretboard as possible, which will result is narrower vibration spread perpendicular to the board, and therefore less buzzing, and no slapping. This does take some time to get a hang off, but eventually becomes second nature.

Another thing you should check is that you have the proper relief in the neck (i.e. the truss rod is adjusted properly) - a little bit of relief can help alleviate a lot of buzzing without raising the action, but too much relief can also be problematic.

Electrics are usually spec'd with low action, since they are played amplified and do not require as much force to get loud sound out of as acoustics do. Low action guitar is best played with light touch.

If you prefer to play harder, you can raise the action to reduce some of the buzzing and slapping (as you said, you might have to raise it at the nut as well, depending on how much action you have on the 1st fret), or you can play more distorted or muted styles which can mask a lot of it, and you can also incorporate those slapping/buzzing sounds into your style, as some people do with acoustic guitars.