You can book through either carrier, although you will need native flight numbers in order to apply PlusPoints -- i.e., if your flight is operated by LH, you need a LH flight number, not a UA flight number, to upgrade it. You request the upgrade online, with United.

Make sure to check the eligible fare classes carefully -- the least expensive economy fares, as well as the least expensive PE fare class, are not upgradeable. You can find more details here: https://www.united.com/ual/en/us/fly...Lufthansa.html

Anecdotally, your odds of clearing LH upgrades are probably better than clearing UA upgrades, especially if, like most people, you're not a Global Services member.