Titletown Brewing of Green Bay commonly did a "double mash" for their very high gravity beers. However, their method was to simply mash and sparge normally, but stop collecting runnings when the wort drops to a predetermined value (the following lower gravity wort was given away to homebrew club members and the floor drain). This is done twice to collect the total needed volume.
There is no reason why mashing with strike-wort wouldn't work. There's just no benefit to it. The enzymes present in the mash would be denatured by the time you went to reuse it, so its diastatic power is equivalent to water. There are several detractors including: possible negative effect on mash pH, higher gravity wort remaining trapped in grain bed, and having to reheat the wort before doubling back in.