At a guess, I'm thinking that it might be a root grafted variety that you have there, and the regrowth may be the result of vigorous shoots coming up from the root-stock.
It is possible to coppice them but not usually recommended, as you tend to get weak sappy growth (water shoots), as opposed to compact foliage with plenty of flowers.
Prune decidouous trees after flowering from mid Summer to early Autumn.
Magnolia grandiflora is the main evergreen variety grown, which flowers later, so again after flowering but before the frosts!