great resource for the company, I would probably want to see if I could extend that another week and find out if there was wiggle room for them. If not I would respect that and let them work the week out. I would plan an exit strategy in order to reassign labor to other key workers with the help of the person leaving.

Kicking them out because they refused to give two week notice is IMO short sited and doesn't help the bottom line unless that employee is already a problem. I totally get having them walk the same day if there presence doesn't shift the needle in terms of profitability. Hiring and training costs money and the more time it takes the more difficult it becomes. So I better imo have a good reason for creating a hole in my labor pool like that. Even for a few days.

It isn't a given that my company is going to always have the optimal number of people working there given the demand. It also isn't a given that my company will always be the most attractive offer for each of my employees. This is one of the many challenges facing hiring managers each and every day.

Personally I think the OP left something out of the story in regards to this employee and their performance historically. IMO that is required to even be able to answer the question if it was the wisest choice.