First, don't cut anything!!! You have a typical white phalaenopsis, and
when the flowers are done, just leave the plant and it will grow new
roots and leaves. There is no dormancy period as far as I know for
phals in the true sense of the word.

Another thing you have to know about phals is that leaves grow from the
crown, and they are monopodial. If you cut it in half down the centre,
unless you get side shoots, you've just KILLED your plant because you
cut through the crown. Cutting the plant in half is not good.....as
phals don't propagate that way. If it's big enough, SOMETIMES you can
divide it in half (top and bottom) and the bottom may give plantlets
from the side since it has a root system, and the top that still has the
crown but no roots will grow new leaves, and the sides will send out
roots. But judgeing by the pic, you probably shouldn't do that. Even
with a very very healthy plant, doing that is not a smart idea. It will
set the plant back a bit even IF it works, and there are much easier
ways to get keiki (offshoots).

Who suggested this cutting gig??? There are culture hints for phals on
the net, I suggested reading a few of those instead. Also, the side
branch you speak of is not of your plant, but of the flower spike, and
cutting that will not give you new plantlets. Occasionally, keikis will
form from flower spikes, but you have to leave them attached to the
plant for that to happen.

Cheers,
Xi