How do I bring back the middle of a video clip I cut?

you can also roll the clip on your timeline. Place your cursor at the end of the clip, your cursor will turn to a red like cursor, then you can roll your clip and reveal back whatever you have removed

Assuming you haven't deleted the raw media itself off of your computer, then sure! Behold the glory of non-destructive, non-linear editing!

Just open the project again, find the raw clip in your project panel, and there you go! You can do whatever you want with it there. View, re-edit, export, etc. Alternatively you can navigate to the clip from a sequence by selecting it, putting your playhead over it, and pressing F (Match Frame) which will queue up that raw media in source at the used range.

Like I said, contingent on you keeping the source media. If you deleted the raw media, the project file alone would not be enough to recover the clip.