I always hate to chime in where I think there have been several quality answers but I think one thing everyone is missing here is that the main issue you are facing is communication based. The employee was fine communicating a threat to you but since hasn't proactively worked with you to create a directional movement and stability, causing you to feel held hostage. As a leader you have a few things to think about. The first one is that no matter how great of an engineer this employee is, if they can't communicate, long term employee retention almost has no chance.

If the employee is trying to leverage better offers from you, they are going to do that in the future proactively if you easily cave to it.

The best thing you can do is become a peer to the employee to try to create open communication and express your concerns. Based on the timeline there is a good chance the other offer fell through and the employee feels stupid but is still looking.

Once you open up communication by explaining to the employee that your goal is to better understand their needs, try to get feedback on the following questions:
What was most attractive about the offer you received? (What is motivating the move)
Can you provide me a copy of the offer so we can refine our compensation or career path strategy with you?
If you are unsuccessful in those, a final question:
Are you truly interested in retaining with us?

If that doesn't move the situation along, I would start hiring to replace ASAP and TERMINATE the employee. You can't be held hostage by someone who isn't even willing to communicate with you.