B737 flight crew reported significant control inputs were needed to correct fuel imbalance during initial climb. The flight crew returned to departure airport and completed an overweight landing.

Narrative

I arrived at the aircraft prior to departure with an inoperative right main tank fuel gauge. I was also met by the individual who was doing the refueling. He was in the flight deck; looking at the gauges; and trying to get the Maintenance Technician to help him with the refueling. The Maintainer told me that he had communicated with him how to proceed with the inoperative right gauge. Method 2 or 3 was to be used; and I would wait for the fuel slip prior to departure. My FO (First Officer) and I resumed our preflight; and were ready to depart approximately 20 minutes before scheduled push back. At that time; my FO looked out the right window to see the refueler standing on a ladder under the wing. I walked out on to the ramp to see how it was going. After getting his attention he told me he was having trouble. At the time he had one of the fuel dipsticks out of the bottom of the wing and was examining it. I asked if he was using either method 2 or 3. His response was hard to discern. I did hear him say that the Maintenance Handler had provided him little to no assistance. I told him I would get him back out there. I then called Maintenance at ZZZ again and relayed the issue. I then called Dispatch; and she said she would get Maintenance Operations involved. Approximately 10 minutes later the same Maintenance Tech showed up on the flight deck; and told me he already told the refueler what do. I asked if he had confirmed method 2 or 3 being used. I think he said that he had. I then sat there for another 45 minutes waiting for the fuel slip. The fuel slip arrived about 40 minutes after scheduled departure. It showed the proper amount of fuel on each side with confirmation that it had been dipped properly after refueling. After takeoff; with my FO flying; he experienced a significant aileron input to maintain level wings. I took control of the aircraft during climb to confirm the flight control input to maintain wings level flight. It required 4.0 on the yoke trim scale. We determined that we possibly had a large fuel imbalance; [we advised ATC] for overweight landing; and returned to ZZZ. After deciding not to cross feed with no fuel quantity information for the right side; we configured early for the approach to ensure adequate controllability. The approach and landing went as planned to [Runway] XX and we taxied back to the gate. Upon inspection Maintenance determined that the right wing had not been adequately refueled as stated on the fuel slip. It was estimated that as much as 4000 lbs of fuel was missing from the right wing.The main factor was inadequate knowledge by the Fuel Technician on how to properly use method 2 or 3 for fueling. It is my belief that Maintenance should have provided more assistance in the measuring of the actual quantity in the tank after refueling. Recommend proper training of individuals who refuel aircraft for Company.