Air carrier Captain reported a loss of oil quantity during cruise and believing the oil pressure would drop rapidly; opted to immediately divert to an alternate airport.

Narrative

During cruise; engine stack started blinking white box with low oil quantity. The quantity eventually went to zero; but oil pressure initially was stable. The FO (First Officer) and I began to look for the Low Oil Quantity Checklist failing to remember there is no checklist for that; a fact which always surprises me; as the condition causes a big white flashing box when there are checklists for probably 20 to 30 un-annunciated failures. We began to look for divert locations as a precaution; thinking the oil pressure would drop rapidly. We chose ZZZ; as we had a tailwind. We [requested priority handling]. In retrospect if I had gotten Dispatch in the loop faster ZZZ1 might have been a better option. I additionally reduced thrust on the #2 in the hopes of limping it to ZZZ. We began planning for ZZZ. I requested lower altitude; which unfortunately made us lose our tailwind. The FO did an outstanding job staying focused and on task throughout. He accomplished all top of descent tasks quickly and efficiently; set up both courses and frequencies; as well as loading the approach and talking to the FAs (Flight Attendant).The flight attendants did an outstanding job getting the cabin secured as well as realizing our task loading was high. I made a PA to the passengers; informing them of our precautionary divert. As the oil pressure continued to drop; we expected we were going to shut the motor down. We began thinking about a nearest suitable; but quickly realized ZZZ was still our best option. We started the Low Oil Pressure Checklist around 17;000 ft. and shut the motor down between 15 and 10. It was an uneventful flaps 15 landing with a normal taxi in. Brake cooling said normal taxi in. Overall; I was satisfied with our performance. FO performed pilot monitoring task at the highest level; and I can't say enough good things about him. For me personally; I became a little fixated on the zero-oil quantity; thinking I would have just seconds to identify and shut the motor down; before a possible uncontainable failure or grinding it down. This somewhat degraded my focus on the checklist and planning with Dispatch. Dispatch's performance was outstanding as well; as they timely sent divert plans and suggestions.