Lay a piece of scrap underlayment and flooring against a door frame to use as a guide and cut just an inch deep until you can remove the base of the frame. Repeat on the other side of the door. Then cut the jamb using your guides. Make these three horizontal cuts at the base of each side of the door frame, being careful to avoid cutting the framing behind them.
Lay the vapor barrier and felt or cork in both rooms through the door. Follow with the thin foam padding, leaving 1/2 inch around the edges for expansion. Tape the edges under the door frames in the same manner as other edges.
Begin laying wood along the longest wall in the room. If this wall is the one the door is on, start on the end of that wall and work toward the door. If the longest wall is perpendicular to the wall the door is on, start on those walls in both rooms at once and work toward the door.
Notch the pieces of wood to fit the doorway, leaving 1/2-inch expansion ease all the way around the door underneath the door frame.
Tilt flooring planks into the lapped edges under door frames or across thresholds. Make an upside-down V with two to ease under the opposite side of the door if the flooring is installed across the threshold.