Since you are going to cut the sod, choose a hot, dry day to do the work (with a couple more in the forecast), cut it, set it aside, rototill the bared plot, adding soil amendments such as compost, manures and  mineral powders as indicated, then replace the sod-- upside down -- and walk away for a few days, letting the upended roots dry out.  Then cover the area with several (10-15) layers of newspaper (most newspapers now use biodegradable soy inks) and begin the lasagna layering. Big chunks of sod don't compost well unless you plan to "pile & forget" but if flipped, dried and buried, the earthworms will have a field day.

I'm not a lasagna gardener, but it is a valid method to skip much labor.