Which is the most recommended way for cycling from Warsaw to Bialostock on a touring
Wylaw there are no recommended E or A roads in PL for cyclists, I have seen enough cyclists dead at the side of the road in PL over the years, to warn anyone NOT to cycle on main roads here.
with appropriate road conditions?
Take your bike by train and go to areas were there is limited traffic like the Tatras or around the Beskids.