B737 Captain reported engine low oil pressure indication during departure climb. Flight crew performed an in flight engine shutdown and return to departure airport.

Narrative

I was the pilot flying on the ZZZZZ departure passing 8;000 for 16;000 ft. when the FO(pilot monitoring) noticed an amber No. 2 ENG pressure alert. We then saw the No. 2 ENG pressure decreasing quickly and turning to a red alert. We received clearance to level off at 10;000 ft. to run the QRH checklist. The FO now became the pilot flying as I ran the ENGINE LOW OIL PRESSURE Checklist. This checklist then took us to the Engine Failure or Shutdown checklist which had us shutdown the No. 2 Engine. The FO [requested priority handling] with ATC and flew vectors west of the airport descending down to 5;000 ft. I finished that checklist; and then the One Engine Inoperative Landing checklist and additional checklist. I briefed the approach to runway XX and then swapped back to the pilot flying. We set up for a 15 nm final and landed overweight around 168;000 lbs. The landing was smooth; and we rolled out to the taxiway 1 clearing the runway there. ARFF (Airport Rescue and Firefighting) was waiting and we asked for and received a 360 degree inspection. The inspection was good; and they followed us to the gate uneventfully. The post flight inspection showed oil around the engine nacelle and a small puddle on the ground. The FO did a great job as we had solid teamwork to get through the checklists and be ready for the approach. This was my first flight since recurrent the week before. After shooting what felt like a dozen Cat III single engine approaches in the simulator this event in the aircraft was almost second nature.