Air Carrier Captain on a post maintenance test flight reported smoke and fumes filling the flight deck from the sidewall. The pilots established priority handling with ATC and returned to the origin airport. The Captain indicated the events were challenging while wearing the oxygen mask.

Narrative

Our flight was a maintenance test flight being performed following the completion of a maintenance check. Shortly after leveling at FL310 and while accelerating to .78M we all noticed a burning smell followed immediately by observance of smoke emanating from right cockpit sidewall behind First Officer's seat. We all immediately donned O2 mask and I (as pilot flying (PF)) initiated left turn back to ZZZ. [We requested priority handling] and then FO and third pilot (Flight Test Engineer) began working the problem with the Smoke; Fumes; or AVIONICS SMOKE checklist. The problem seemed to resolve shortly after the first 6 steps of the checklist were performed. Smoke was no longer visible and odor was not detectable. [Priority] status was maintained in order to keep priority handling with ATC and RTB continued at best forward speed. Shortly after descending through 10;000 ft. the situation reoccurred this time with heavier smoke emanating from cockpit sidewalls on both sides. At this point we were being vectored onto downwind for XXR ZZZ. We requested immediate vectors to XXR and were offered Runway XX which we accepted as it was off our nose and less distance. Some S turns were performed on short final to lose altitude and airspeed followed by a full stop landing on Runway XX. By then smoke had once again dissipated. We opened cockpit windows and ran first 6 steps of Evacuation checklist stopping before Fire switches (Engines & APU) ...Push as engines had shut down normally and APU was never started. Fire Crew and Tower both reported no evidence of fire so we remained on Aircraft with windows open until Tug and Airstairs were brought to aircraft. We did leave batteries on for the purpose of radio communication with Tower. No additional info.The cause is unknown at the moment. Our recommendation for ourselves as well as other aircrews in the industry is to further train communications with O2 mask on. Things to emphasize would include volume levels of cockpit speakers; knowing in advance whether or not you want to remove cockpit headset and glasses; proper use of intercom transmit switches; etc. The amount of background noise generated by breathing O2 is highly distracting and could be significantly mitigated by more training and comfort with the use of the O2 mask. Having just come out of the Covid pandemic I suspect that emphasis on this type of training has probably waned industry wide.