I'll present the technique alongside such an example. Proof-of-concept but also something emotional: Trying to refactor an Aleppo soap to smell less prominently of laurel oil. (Sacrilege! 
)
Some time ago I've bought some Aleppo soap with 30% laurel (couldn't be bothered to buy laurel oil to make it myself, lol), and felt like this was the one to get my fingers dirty with. I chopped up coarsely (we'll see the revenge for my laziness soon).
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Then I added a molar excess of citric acid (25 g for 100 g of soap):
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The rule of thumb for dosage: Use a soap calculator and enter the recipe. Each 4 g of NaOH will need 7 g of citric acid, plus some more to be on the safe side. Excess acid really doesn't matter. I've used citric acid because I have it at hand. Acetic acid (vinegar) would work just as well (just smell a bit), an so do hydrochloric or sulfuric acid if you have access to it and are willing to deal with the extra safety precautions.
Next: top up with some water.
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The acid will dissolve into the water and start its action: Grab the sodium from the soap (sodium salts of fatty acids) and exchange it for protons. This liberates the free fatty acids, and leaves anything water-soluble (sodium citrate, glycerol, salt) in the water phase.
I let it stand for several days, when I noticed that there are large chunks of unreacted soap left.
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Heat, thorough stirring, time, and the threat of the stick blender did help, though.
Pro tip to my former self: Don't shy away from a cheese grater. It pays off. 
For Aleppo soap, FFAs are mostly oleic, linoleic and lauric acid, so the FFA layer is expected to be a free-flowing liquid, at least above the melting temperature of lauric acid (44°C). Recipes with significant amounts of palmitic and stearic acid (Mp 63°C/69°C) will need much more conviction to fully separate.
Anyhow. Once there is a reasonable separation into a sticky, murky FFA layer, and a turbid, brownish-green aqueous layer, it's time to separate them. Not unlike rendering tallow etc., cold comes to a help for us – the FFA layer will solidify in the fridge (oleic acid Mp 13°C) to a waxy mass that we can easily scoop off the water/waste layer.
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100 g of store-bought Aleppo soap left me with 76 g crude FFA, still containing some water, but hardly any glycerol, sodium, or water-soluble side-products of laurel oil saponification.
