recover/restore files with specific extension from Time Machine with OSX
It seems like you will have to use the Terminal:


Mount your Time Machine disk. It will be located in /Volumes/myTMdisk/.
Start the Terminal
Type rsync -av --dry-run --include="*.gif" --include="*/" --exclude="*" /Volumes/myTMdisk/Backups.backupdb/myMac/Latest/myDisk/Users/me/ /Users/me/Desktop/allmygifs/ 


Replace myTMdisk, myMac, myDisk, me, and /Users/me/Desktop/allmygifs/ with the appropriate values. 
You can use the tab key  while typing to auto-complete your path.
If your pathname contains whitespaces or special characters, put it in quotes, e.g. rsync -av --dry-run --include="*.gif" --include="*/" --exclude="*" "/Volumes/Seagate Backup Plus Drive/Backups.backupdb/david914’s MacBook Air/2015-08-30-221221/Macintosh HD/Users/david914/" "/Users/david914/Desktop/allmygifs/"
If you want to restore the files into the original locations, use rsync -av --dry-run --include="*.gif" --include="*/" --exclude="*" /Volumes/myTMdisk/Backups.backupdb/myMac/Latest/myDisk/Users/me/ /Users/me/. I would recommend another backup before doing that. Then replace Latest with the respective timestamp.
In case you get file permission errors (e.g. Permission denied and code 23 error as described here), you may have to use sudo in front of rsync. But now you will really need to backup first. Ideally backup to a different disk or make another copy of your TM disk.

Once you are sure everything works, remove the --dry-run argument after rsync -av to actually write files.
Enjoy!