Air carrier Captain reported a smoke event in the aft galley during initial climb. Flight crew returned to the departure airport and completed a safe landing.

Narrative

After takeoff from Runway XXC at [Taxiway] XX around 800 AFL we received a call from the number 2 Flight Attendant that smoke was in the aft galley. I was the PM (Pilot Monitoring) and told the number 2 FA (Flight Attendant) we would be returning for landing at ZZZ. I [advised ATC] with Tower. Tower told us to switch to Departure and I [requested priority handling] again with Departure. We were asked what altitude we would like; and continued the climb to 5000 ft. while turning to a downwind. I pulled out the QRC and began to read it which led me to the QRH. Seeing the length of the checklist and how close we were to landing; I used my Captain's judgement and emergency authority to put it away and concentrated on getting the jet back onto the ground. After putting away the checklist; I called the FA's to see how they were doing and told them we would be returning to ZZZ with ARFF standing by. After this I made a PA to the passengers telling them about the smoke in the cabin and that we would be returning to ZZZ with safety vehicles standing by. I then contacted Dispatch through ACARS and stated smoke in cabin and requesting ARFF. We were on base and I determined we needed to do an overweight landing; so I pulled out that checklist and had no time to run it because we were turning a 5 mile final. I told the PF (Pilot Flying) to land soft and I would be writing down the V/S; airspeed and weight. After taking care of everything we ran the Before Landing Checklist while on final and had everything completed before our stable call at 1000 ft. When we landed we rolled onto a taxiway and stopped. After stopping I informed the passengers we would be inspected by the safety vehicles; followed by going to the gate. When ARFF arrived; they determined we were clear and had no heat coming off the jet. They followed us to the gate and boarded the plane after we parked. They gave us the clear that we were safe to deplane; however; by the time they got on the plane the electrical smell and smoke had dissipated. I wrote up the airplane and overweight landing protocols from the FOM.Having smoke in the cabin as an only indication; I didn't know If we had a fire or just smoke from a bleed system. I know how dangerous these fumes can be so that is why I elected to return as fast as possible and did not hold to finish up any checklist. These are times when you have to use good judgement and captain's authority.