Why are 2 electrons transported from photosystem II at the same time? I only know that electrons are captured by the primary electron acceptor and then go through the electron transport chain, ultimately ending up at photosystem I. But why do 2 leave photosystem II in the first place? I thought the oxygen-evolving complex only replenishes the electron hole one-by-one?

So refer to the following diagram,

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In the photosystem II complex, water is decomposed into oxygen and protons. That's two electrons liberated from each water molecule. Plastoquinone accepts two protons from the stroma by coupling it to the two electrons it receives from the photosystem complex. In this manner, the protons are transported to the lumen, and the electrons are sent into the electron transport chain. So PQ binds in a QB site in the D1 protein, and it's not released until it's double reduced/protonated. Another PQ binds in it's place when PQH2 leaves the binding site. Source

Short answer: Two electrons are liberated in splitting water, and that mechanism feeds into asymmetric lumen/stroma H+ concentrations and movement of electrons through the ETC.