Air carrier flight crew reported an electrical odor during flight. The flight requested and was provided special handling to a suitable divert airport.

Narrative

We were flying ZZZ1-ZZZ2. First Officer (FO) was the pilot flying and I was pilot monitoring (PM). The weather was VMC. At top of cruise I noticed a faint electrical odor in the galley area. My FO also agreed he smelled a faint odor. At the time it was not enough to cause alarm but we started the process of trying to determine the source. After approximately 45 minutes we ran the non-alert Smoke/Fire/Fumes checklist which had us remove power from the galley area via the Cab Bus Switch. Over the next 5 minutes the odor became more pronounced. Not being able to determine the source; we decided to divert. We donned O2; informed ATC of our intention to divert to ZZZ. They gave us vectors and a descent. I sent a divert message via ACARs and set up the ILS Y XXL to ZZZ. We [requested priority handling] and reported our fuel and souls to ATC. At this time I decided to become pilot flying (PF) and the FO became PM. He obtained landing data for XXL where we realized XXL had been shortened to 7000. We were able to get MAX numbers so we continued. We had ZZZ in sight from 25 miles out but we were unable to tune the ILS. The ILS had been NOTAMd out for XXL. We asked Tower if they could turn on the ILS and they said it was unavailable. We had guidance from the NAV/PROF as well as the VASI for XXL. We landed uneventfully. After landing we determined we were unable to 180 on the 150 wide runway or taxi off on 1 due to weight restrictions. We made the decision to run the evacuation checklist and exit the aircraft on the runway. I blew the slide. The galley area now had visible haze and very pronounced electrical odor. Once the slide was deployed I exited the aircraft and my FO followed behind. The slide is very steep and very fast. My FO injured his hip during his dismount at the bottom. The whole experience was very slow developing. I found that it was hard to make the divert decision due to the slow onset with no associated indications. Once the divert decision was made the process becomes much tougher with the oxygen mask and goggles. My FO and I had outstanding CRM throughout the entire process. That open loop decision making enabled us to keep a very high level of safety throughout the flight. We have flown together several times. We had been using the technique of having divert fields in the fix page. At the time of the divert we had ZZZ3 and ZZZ and decided upon ZZZ due to familiarity with the airfield. Reading the NOTAMs prior to the divert decision would have helped immensely. Obviously this would need to be done well in advance of the actual divert.