Take a look at Wobdees posts about the Schmitz decoction method.
Here's what will happen if thick mash is pulled, boiled and added back to the main mash in order to reach mash out temp. The thick mash contains hard starch that is stuck in the tips of the husk. The temperature at which hard starch begins to burst and enters into solution is around 168F. The boiling of the mash causes the starch to burst. The starch laden mash is dumped into the main mash to reach mash out temp. Enzymatic action has ceased due to denaturing by heat. Excess starch carry over occurs, reducing stability and shelf life of the beer. Mash pH needs to be considered before mash is boiled. Usually, the thinnest part of the mash is removed, boiled and added back to the main mash to reach mash out temp..
Part of the decoction thing is to extract as much as possible from the malt. Starch is one of the things extracted. Throughout the process, the starch laden mash is rested at various temperatures to activate certain enzymes. The action of the enzymes reduces the excess starch into sugars. This is a reason why a tri-decoction will produce a higher gravity wort than a wort drawn from the English method.
You can cool a decoction. Unless, you are dead set in doing a decoction for mash out, maybe consider this. Do an hot water infusion to reach a beta rest temp of the main mash. Soon as the mash temp is stable. Pull a thick decoction. Heat the decoction to 153F and rest for 10. Then, take it to a boil. Boil the mash for at least 10 minutes. Realize, it takes longer than 20 minutes of boiling to begin forming melanoidin. Cool the mash to a temp so that when it is added back into the main mash, the mash temp will be in the alpha range. The process will add a little more character to the beer. It might take a while to dial in the timing....Anyway, just some ideas....Good luck, brew on!