I have a Time Machine backup (of personal files only) made from my MacBook Pro provided by my previous employer. Now I have a new job and a new (employer-provided) MacBook Pro. Can I somehow restore files from that backup onto my new machine using the Time Machine UI?
The backup they gave you; is it just files or is it the
The Time Machine backup you have is "tied" to the old computer. You do have a couple options to restore the data however:
Link Time Machine To Your New Computer
Check out this other question on SuperUser: 4906 for how to migrate your Time Machine backup to work on your new computer as well.
Browse Other Time Machine Disks
Hold the option key and click on the Time Machine Menu Extra and choose "Browse Other Time Machine Disks".