How you break an emulsion depends somewhat on whether the continuous phase is aqueous or oily, oil droplets in water or water droplets in oil. I suspect you have oil droplets.
These are just a few methods I would try (in addition to those already mentioned earlier by users Sobachatina and fyrepenguine) if I were faced with this, but I have not tested them on your exact combination.
Adding warm or hot salt water (0.5% by weight salt in water) into it and stir
Thermal treatment:
Extreme heat cycles (freezing and heating)
Particle charge disruption methods (all different methods not steps in a single method) :
- Adding vinegar and stir
- Dissolving agar or gelatin hot water and add to emulsion, stir and let it
settle and then separate the oil from the gel (1-2% by weight)
- Add salt (0.5% steps by weight) to gel solution above and stir into emulsion
G-force:
If you have a centrifuge, there should be enough density difference between the two phases to achieve a complete separation.
Carotene has a strong preference to stay in the oil while salt, vinegar strongly prefer water.
As an aside, with aggressive agitation and when subjected to high shear, you tend to end up with a very stable emulsion. This is why my preferred way of this type of 2-phase liquid-liquid extraction is to use gentle stirring over more time rather than using a blender. Even periodic shaking is preferable. Diffusion across the interface will allow the extraction to happen with time. If you use a thermometer to monitor the temperature of the contents inside most kitchen blender, even in high speed mode, there is no heating effect from friction (except for very high power ones as pointed out in comment below), but You are very likely to create electrically charged droplets which is great if you are after a stable emulsion. Very high speed blenders capable of heating by cavitation can make the emulsion more stable than less stable. There is also the likelihood of aeration to accelerate enzymatic browning.
It is too easy to just focus on the best way to extract (small particles for high surface area and short diffusion path) in the least amount of time, and lose sight of the entire process which still requires separating the oil phase from the water phase. Extraction between two very mobile phases is relatively painless, but emulsion breaking can be painful, and you want to avoid doing one step really well only to create a hard problem for later.
In a very crude experiment today, I found that starting with pulped pressure-steamed carrot, using a gold coffee filter cone and repeatedly dripped clarified butter through with patience worked very well, no emulsion (just a layer of water phase below to decant away). In fact, if you really want efficiency, retain some carotene butter for next time, start the extraction with fresh pulp and melted carotene butter in the first couple of runs, then use pure clarified butter for subsequent once. (Counter-current extraction technique). Strictly speaking it is no longer liq-liq extraction but leaching but the concentration gradient principle is the same.
PS
This is another technique that is much more sensible/workable in a home kitchen.
Shred the carrots, finer is better but coarse shredding works well enough. You can optionally blanch this if you are concerned about browning.
Put into a bag with hot clarified butter at above 75C, expel as much air as you can or vacuum the bag before sealing. Sous vide at 75C or use a hot water bath (well below boiling) for 3 hours or longer. Decant off the now orange coloured oil phase. You can keep the carrot pulp to make crisps or discard it. If you are obsessed with extraction efficiency, cook it for a day, but I suspect numerically you would not get to the same degree as the juice/emulsion method, but then this is practically just a single extraction-separation step without any emulsion headache.
