SATs lightning rewards to self custody / cold storage

If I'm earning sats and can send them through lightning, what's the best way to move those sats to self custody / even cold storage? What's the min amount that would make sense to move from lightning to on chain and how can I only pay network fees (avoiding 3rd party fees)? I've looked into a few lightning wallets for Android in the US and didn't see a consensus on what to use.