To be formalized eventually. For now, here's a Slack post from Shyam.
NEB says to phosphorylate by adding 300 pmol 5′ termini with 500 nmol ATP, 1 µL T4 PNK, 1× PNK buffer, in a 50 µL rxn. PNK has full activity in ligase buffer, which also supplies sufficient ATP. We don't keep ATP aliquots around anyways to use PNK buffer.
I anneal pairs of oligos first to ensure their 5′ termini are exposed for later phosphorylation, though in the literature, most people individually phosphorylate oligos first, assuming the 5′ ends will be exposed in whatever 2° structure the oligo adopts at 37°. Then they anneal the phospho-oligos. I don't make assumptions. Assumptions invite failure.
Anneal oligo pairs in water, or TE as I realized ought to be better due to buffering without Mg, at something like 2.5 µM oligo in something like 10 µL. That's 0.25 µL of each 100 µM oligo (=25 pmol each oligo = 50 pmol total termini). Run thermocycler anneal program. The long 45 min one is thought to be "better" than the short 20 min one, but in practice, the short one has been fine for me in making things with intended strong secondary structure. Unnecessary high temp exposure to oligos ought to be minimized anyhow.
Then phosphorylate by spiking in 10× T4 ligase buffer to a final 1× and then minimal amount of T4 PNK. 50 pmol termini can be phosphorylated by 1/6 µL T4 PNK by the previous rule. Round up to ¼ µL since that's the minimal pipettable, and you can't make a master mix with it and just concentrated buffer. If you want to make a master mix, make at most a 2× master mix of buffer and minimal enzyme and add it to the presumably several annealing reactions (10 µL MM + 10 µL anneal). Incubate 37° ≥30 min.
Now the phosphorylated, annealed ds-oligo product should be at a bit less than the initial 2.5 µM (due to the small added volumes, or 1.25 µM if you added a 2× master mix's volume). A 1:25 dilution into water or TE would put this product just under two-fold (~100 nM) my prescribed concentration for Golden Gates (50 nM). Not "luckily" (I've obviously designed it to be this way), this can conveniently be done by adding 240 µL water to the top of each rxn. The PCR tube cannot hold any more volume. Any more concentrated the previous reactions' oligos, and you'd need another tube to make a dilution. That or just use <0.5 µL when you use it, or end up using >25 fmol in the GG with no issue. What I thus give you is a one-pot, sequential annealing-phosphorylation reaction, more efficient by its use of three less pots/tips per rxn and half the enzyme than what my great predecessors taught me.
As you can see, I'm able to make something like mixing two complimentary oligos into a mind-bending exercise to understand.