2006 GT mustang stutters or hesitates from idle but then is fine. problem goes away after warming up
Title says most of the problem. It's like it runs rough for a split second the moment you touch the gas and rev it from idle. It can be replicated while driving only if it hits idle again. Otherwise switching gears while not fully idle doesn't reproduce it.
I usually dive it 10 highway miles to work every day - by the time I get there the problem is almost always completely undetectable.
The problem showed up the same time that an engine code popped up and at AutoZone I pulled a P2004 but the engine light went off a day later. Problem persisted. Took it to a mechanic (who ignored all of the above except "customer reports car running rough") and pulled a cylinder #2 misfire code to which they decided to replace all plugs and coils. Problem was not resolved. Cylinder #2 misfire code still present. Yep - don't think i needed those plugs or coils.
Going back a bit further in history the car did have its intake manifold off and gasket replaced in order to replace the heater hoses under the intake manifold but this was almost 10K miles ago at this point and long before this problem popped up.
I checked out the air filter - clean. Also got some mass air flow cleaner and cleaned the MAF. No luck there.
I'm not a mechanic, but thinking my next steps should be:
1) inspect in the throttle body and give that a cleaning
2) replace injector #2 and see if that helps - this seems pretty straight forward
3) replace the fuel filter (It's possible this is ~8 years 30K miles old - which I realize is old. I can't find a record of it being replaced sooner than that). This is maybe a bit more messy
Just wondering if anyone has better ideas - thanks!
Title says most of the problem.
It's not supposed to. The title line is intended for a short description. (Title edited.)

Inspect in the throttle body and give that a cleaning
You might want to clean the throttle body and MAF sensor.
https://www.youtube.com/watch?v=NWCKvU2FmDc
replace injector #2 and see if that helps - this seems pretty straight forward
Why not swap it with another injector on the engine and see if the problem travels with it? You might also want to do a compression check or leakdown test on that cylinder.
replace the fuel filter (It's possible this is ~8 years 30K miles old
Not a bad idea but you might want to also check fuel pressure.
A full diagnostic scan including live data or any freeze-frame data available might shed some light on the problem. If you can record the data when the problem happens for playback and analysis later that may help.