To answer the question of how I clean, first check out the picture of my system (which hopefully shows up, it's a pic when I'm doing a fly sparge).
With higher grain bills (Belgian beers) I was struggling with stuck mashes. Because you need to keep the electric coil fully submerged in the BK, you basically get handcuffed on your mash thickness in the MT. I did rice hulls and other tricks, but I still struggled with stuck mashes when grain bill was high. I also really couldn't stand how long the temperature control (in auto) took to hit setpoint. So besides the handcuff that I mentioned, you also have a greater volume of water that you need to heat in the BK, so I decided to get rid of the lag time (due to the volume) by controlling my mash temp with the RIMS rocket, plus I can run thinner mashes. I also wanted to use the hopback feature, which I just did yesterday on an IPA using homegrown hops.
For my first batch with the RIMS rocket, the temperature control was very cyclical. Next batch I spent a hour or so trying to figure out the temperature controller settings before I brewed, I did some digging online and found a delta temperature controller user manual that looked VERY similar to the Blichmann temp controller. I noticed the temp controller was programmed to ONLY use gain, no Integral or Derivitive (all P, no I or D). Temperature control systems are one of the few loops the typically use P, I, and D (most loops only use P and I). I didn't have they instrumentation and control loop tuning toys I had when I was an engineer, so I relied on the auto tune feature (look at your temperature controller, see on the lower left where you see AT...that lights when the controller is performing an auto tune). Very snappy controls after auto tuning, without overshoot as the other attached picture shows (hopefully you can see the pic).
Onto the cleaning question. After I sparge, I unplug the RIMS rocket and relocate the plug into the BK. (My brew day is slower than it could be if I spent $500 on another TOP controller, I could boil while I sparge, but for an additional 30+ mins on brew day (=1 more beer to drink) I'm cool with a slightly longer brew day.)
I put the HLT away, I dump the MT and hose out as others mentioned. I'll pull out the RIMS rocket (real easy to clean manually) and replace that with the HX. I put PBW in the MT, then pump room temp PBW from MT through the pump, HX, flow meter (I have the full TOP), RTD and back to the MT. Run PBW 15 mins, dump PBW into a bucket for next brew day, water rinse the system, then run Star San for 15 mins. I dump the star san into another bucket, loosen up fittings to drain starsan from HX, pump, etc...then you're ready for chilling. I whirlpool when I chill my wort, when I first turn on the pump, I'll pitch a few hundered mls of wort, I figure it probably has a good bit of StarSan that I didn't drain.
In using the TOP to whirlpool, I find it to be pretty handy because you can see your wort temp leaving the HX on the RTD, and you can see the temp in the BK. My city water temp is real warm (mid 70's) so I use my former immersion chiller to pre-chill the city water in two 20 lb bags of ice (ice is in a cooler).
One interesting thing that I noticed now that I don't use the BK for "K-RIMS," there is WAY less crud stuck on the BK heating element. My LHBS owner, whom is also a beer judge, thought my beers had a carmalized taste to them, since going to RIMS rocket, he doesn't taste that anymore, FWIW.
Oh, after brew day is run, I'll do the official 160 deg F PBW cycle, water rinse, and acid sanitize.
You now have read a summary of 15 batches of knowledge using the breweasy which I purchased in late May. While the RIMS rocket added more expense to an already pricy system, I'm really happy with the control, flexibility, and efficiencies I'm getting now. The beer of course tastes great, but I thought my beers tasted great when I was an extract brewer. We'll save the AG vs extract debate for another day