The white smoke can only be 2 things, fuel or coolant. As suggested, if it is the EGR cooler you can delete it, but you can also park nose low and pull the EGR valve to see if there is coolant (not moisture from sitting) in the intake manifold and you should have noticed the coolant level dropping, sometimes slowly. This is easy to do even if you only own a small socket set!
If it is fuel, the FICM firing incorrectly can dump raw fuel into the cylinder that will come out the exhaust as white smoke. It could also be an injector (usually black smoke but could be white) or even an exhaust valve not closing. Revving the piss out of an engine with a problem won't fix it and it can make it worse. Take the time to methodically work through the problems I have suggested. Get a solid diagnosis from a different shop or, if you are capable and willing, get a diagnostic tool and try to trouble-shoot it yourself.
So what did they tell you was done? A trained mechanic, using IDS, should have found the problem without it breaking the bank. You need to itemize what the $4k was. If it was one injector on both heads, the bill would be significant, but I don't think any dealership could present a bill that high for what you said was done, but as we all find out, anything is possible.