2006 ford five hundred 3.0 engine.
How the Heck do you remove this and what tool or specialty tool would be best used?

 

The ideal way to remove it would include a split wedge type that goes behind the pulley then a long leg puller to pull on the wedge assy.

I hesitate to recommend a puller that just goes behind the pulley with 2 or 3 legs because that can break the back of the pulley and now you will need to get a replacement.
Might be a good idea to check with an auto supply store that either rents or sells the correct tool. Cheaper to buy the tool and reuse the old pulley versus have to fight to get the old one off then still have to buy a new pulley,,,,,,,,if you can get one ! I believe they are vehicle specific because the crank sensor timing teeth are also on the back of that pulley.