With the projects I've worked on so far, I tend to cut everything at once (like you, I make large messes pulling out all the fabric). Then I usually start on the piece of a quilt that I'm most excited about, then finish the whole section(s). Then I usually work row to row (ie, the one I'm doing now has a complicated vertical strip, then like 12 rows of 4 blocks, so once I had the complicated vertical strop pieced, I've been doing each strip of rows in stages. Do the 4 blocks, attach the four blocks, move to the next row)