How can I restore Time Machine backup to an external SSD on a Mid 2012 MacBook Pro?

When I try to do a restore to the external drive, utilities does not see the external drive it only lets me choose the hard drive to restore to.

That is correct. To accomplish what you seek, you must first install macOS on that external SSD. Boot from it, which will launch a setup assistant. When asked if you want to transfer content from another Mac or its Time Machine backup, that's what you will want to do.

• It really does not matter which option you choose, but given the fact the existing startup disk has become intolerably slow you will probably want to choose its TM backup instead.
• The macOS version on that SSD must be equal to or later than the macOS version used to create that Time Machine backup.