It could still be the EGR. When you step on the gas, in normal operation it creates a vacuum that signals the EGR to allow exhaust gas into the fuel mixture. It’s done to cool the combustion process in order prevent to much nitrogen containing pollutants to be formed during accelerations. Usually this is disabled if the engine is cold. Does this problem occur even if the engine is cold?
One thing you might ask your mechanic is if he can temporarily disable the EGR to see if it fixes the problem. If so, you know you just need a new EGR (and probably the EGR ports cleaned out of carbon deposits.)
There’s other stuff that could cause this same symptom too. With no diagnostic codes current or pending, if the problem isn’t the EGR, its going to be tougher to diagnose. Anything allowing unmetered air into the intake manifold could produce this symptom; i.e. a brake booster diaphragm split, faulty PCV, etc. Usually that would produce a “lean” code though.
Here’s what I’d do if I had this problem on my Corolla. Besides reading out the codes and a visual inspection of the workings, especially looking for air leaks, I’d replace the EGR and PCV with new ones, and clean out all the ports. I’d clean the throttle body of deposits and make sure the IAC was clean and working correctly. If there was any routine engine maintenance that was due (according to the maintenance schedule) , like new air filter, new fuel filter, new plugs, valve clearance measurements, timing belt, idle rpm setting, etc , I’d do that.