That style(Victorian) does not have an aerator in the spigot itself, so we can rule that part out.
You can, however still have a screen in the connectors of the supply lines.
Does it only happen on your cold tap on that sink or does the hot behave oddly as well?
That tap is very easy to rebuild. My guess at this point is that you have some corrosion or deposits in the faucet causing the behavior. If snaking something like a zip tie or wire into the mouth doesn't result in removal of anything, I'd consider pulling the tap apart(after shutting the water off and releasing the pressure). There's really only a couple rubber seals in it that are the consumables, the rest should be able to be reused after cleaning.