Never done a door, but have done lots of cabinets and they have doors.

Foam brush, argh.
Getting poly on with any brush is difficult without dripping and globbing;
probably impossible with foam.
I use wipe on poly. It is thin, so you need twice as many applications, but
they each take half as long, so you are even there. The important thing
is that it never drips, globs, or skips.
I use rags that come 100/bag; cheaper than foam brushes even.