After a brief bit of research, it sounds like chaumian ecash is a means to scale Lightning (at least in the context of protocols like Cashu).

Users exchange pieces of data that represent ecash, which in turn represents Bitcoin.

Custodial mints are responsible for verifying transactions and holding the underlying Bitcoin, that backs all of the ecash traded by users of that mint.

Users have a relationship with one or more mints, at zero cost (unlike the costs associated with opening lightning channels), and can interact with users of other mints, by virtue of the fact that mints are interconnected, via the Lightning Network.

I am keen to learn more about the nature of protocols like Cashu, specifically in terms of the benefits and safeguards they can offer users who cannot afford to operate directly on Lightning, due to high on chain fees, technical barriers, or otherwise.

I'm also interested in speculation around who will end up running the mints...