Learn German: Er behandelt das Register vorsichtig, wie wir besprochen haben. - He handles the index carefully, as we discussed.

You are right that wie is usually the question word 'how'. However, it is also used as a subordinating conjunction to refer back to something people already know or previously agreed on. Here, wie wir besprochen haben means 'as we discussed' or 'in the way that we discussed'.

Because wie acts as a subordinating conjunction, does that explain the word order at the end?

Exactly. Since wie introduces a subordinate clause, it kicks the conjugated verb to the very end. That is why the helping verb haben comes right after the past participle besprochen.

Wait, doesn't Register mean a cash register in a store?

That is a very common false friend! A cash register in a shop is die Kasse. In German, das Register refers to an index in a book, a database of official records, or a musical register.

Could I use the normal verb for speaking and say wie wir gesprochen haben?

Using besprochen is much better here. The verb besprechen means 'to discuss' a specific topic, which fits perfectly because they made a plan about the index. Gesprochen just means 'spoken', which does not convey that a specific agreement was made.