exported frames brought into the timeline don't match color

Yes, it is a bug in version 11.

Join those of us reporting it using feedback: Final Cut Pro->Provide Final Cut Pro Feedback...

I am examining this. There might be an anomaly unique to FCP 11. However, comparing still image frame exports to video has never been reliable. This is because the still image has different color space metadata than the video. When ColorSync matches your display profile, it is essentially comparing two different image formats with different color space data.

It's a better technique to export a single video frame from FCP which could be smaller than the original frame size. E.g, if the project is 4k, select a frame then use the I/O and JKL keys select a single frame and export that as 1080p. Then use *that* as the color reference. That small one-frame video file will contain the same NCLC and other metadata as the original timeline clip. Then if you re-import that on the original or another Mac, it should stay the same -- assuming both machines have the same display color profile as seen in System Settings>Displays. This does not consider monitor calibration but that is normally a lesser factor than the metadata and display profile issues.

The above assumes the still frame is being used as a color reference, not as timeline program material. There are cases where the content itself is a JPG, PNG, or TIFF video frame, and you want that to match between two projects or two machines. Let us know which case this is.

Please also state your MacOS version, monitor type, whether it is connected via HDMI, Thunderbolt or a built-in iMac/MacBook Pro monitor, and what the display profile is for that monitor as shown in System Settings>Displays.