There is no one, specific committee in Congress that is the only one whose members can develop expertise on a complex issue. Instead, there are many such committees. I believe that this question should read something like “what type of congressional committee…” The type of congressional committee that allows this is the standing committee.
There are generally said to be three kinds of committees in Congress. One is the conference committee. This type of committee is made up of members from both houses of Congress. It is created specifically to iron out differences between the versions of a bill that have been passed by the Senate and the House. The conference committee gets together long enough to agree on how to make the bills identical. Then it disbands. Another is a select committee. This is a committee that is put together for only a limited time. It is created for a specific purpose and for a limited time. Neither of these committees exists long enough for its members to become expert in a given area of policy.
By contrast, standing committees are essentially immortal. They will continue to exist unless some future Congress votes to disband them. This rarely happens. Therefore, the members of these committees can remain on the committees for long periods of time. This gives them the chance to develop expertise in the area for which the committee is responsible.