Is There A Best Case Scenario Which Minimises Carbon Build Up In EGR Circuit?

It's a waste of time to look at that huge pipe. The cooler is what clogs. Those passages
are small. Same with the intake manifold.
Changing the oil/filter at 5000 miles helps, not driving the Prius like an F1 car helps.
Installing an oil catch can helps. Nothing will stop it, it's a by product of engines.
Highway miles are better for the Prius then city driving.

Using a higher octane than what Toyota calls for is just wasting your money. It's not going to help
the engine at all. Quality of the fuel will.

 

Just clean it. Shouldn't have to be, but Toyota screwed the pooched, and doesn't want to own up.

 

The op said he can't find a shop that wants to do it.

The best approach is to keep your 2011 from burning oil. Oil burned through the combustion chamber creates excessive carbon which clogs up the catalytic converter and will fully clog up the egr cooler.

If it is already burning oil (which is most common after 125,000 miles) the engine needs revised pistons and rings. This problem was acknowledged by Toyota years ago and changed free if oil burning became excessive while still in the powertrain warranty.

Many of us simply view the automatically calculated egr flow on a regular basis:


A good egr flow is typically 18-22.

This comes from the bluetooth phone app "Car Scanner" or other obd2 scanners including Techstream. The car will code if this periodic test fails.

Another easier to achieve egr related maintenance is to have a shop replace the intake manifold which may be the only credible egr flow clogging concern.

If head gasket issues are the real concern, the egr cooler clogging theory is not shared by Toyota, their mechanics and apparently your guy.

If existing stumbles and rough running are your concern, an egr block flow test is easy for your shop to accomplish and can determine if the egr valve is sticking open, a failure mode that sometimes happens and was also covered by Toyota for a period.

Egr block flow test

A revised egr valve is available as well.
Valve Kit With Gaskets
04004-58137 - Toyota Parts Deal