Air carrier Captain reported a fume event shortly after takeoff when a Flight Attendant notified the flight deck of an odor in the cabin. The flight crew requested special handling to return to airport while completing the QRH procedures and executed a safe landing.

Narrative

We had numerous calls to Maintenance and Dispatch via ZZZ1 Radio for maintenance problems and MELs; runway changes; route changes and required fuel. After a very long taxi and one more call to Maintenance concerning an EEC (Electronic Engine Control) MEL; we were cleared for our packs off take off. The FO (First Officer) was the Flying Pilot and after I had set the power manually; due to the MEL; and near 80-100 kts.; I heard a 'ding' and later saw the fwd cabin call light on. We continued the take off and climb out. During climb out; I called the cabin and the A [Flight Attendant] said he smelled an odor and he could not see any smoke. He I looked around and the odor was getting worst. As we continued to climb; he told me that he was feeling lightheaded and the JS (Jumpseater) sitting next to the 1L door felt like she was going to pass out and the Flight Attendant at the 1R door was not feeling well. I told the FO what was going on and we put on our O2 masks and told ATC that we had a smoke/fume problem. We were level at 4000 ft. and I started the smoke fume QRH procedure; and was given a climb and getting information from the A flight attendant. He said the odor could be smelled in the first class cabin and a few passengers were feeling lightheaded or dizzy. I told the FO to tell ATC that we needed to level off and they; ATC asked if we needed to return to the airport and if we [needed priority handling]. I said yes and ATC gave us a lower altitude and heading for the ILS XXL. I told the A that we were returning to ZZZ and we had an [event] and be on the ground in 10 minutes. I made a PA to the passengers and told then we would be returning to ZZZ. I completed the QRH procedure to the point it said don't delay landing to finish the checklist and we setup for the ILS XXL. We got a delay vector to prepare for landing and the FO flew and landed flaps 30 auto brakes max. An over weight landing did occur. We taxied off the runway and emergency vehicles followed us to the gate. We informed Ramp Control to tell the Ground Personal not to get near the tires due to the hot brakes. The cabin door was opened and the Gate Agent could smell the odor and a few of the flight attendants who walked up to the front of the cabin also could smell the odor and felt lightheaded or dizzy or headache. The Flight Attendant Supervisor who came onboard could also small and odor. EMT personnel came on the aircraft and checked out all customers and flight attendants who were not feeling well