The Circle line through Barbican almost fits this description - it was
in open cutting before being rebuilt in tunnel when the Barbican
Centre was built.

Did they lower the level of the tracks when they built the Barbican
Centre? Or did they just use the valuable air space over the tracks in
the cutting?

Neither - the line was completely re-routed, the new route being in
tunnel.

Contrariwise: they just used the air space over the tracks. See, for
example, this history from David Leboff's "London Underground Stations
abc":

# The original station building was a single-storey,
# light-coloured brick structure which was rather plain
# compared with many of the stations constructed at around
# that time. It was severely damaged during World War 2 and
# finally demolished around 1955, along with the standard
# overall roof which extended over the platforms and tracks.
# During the early 1960s, the entrance and a new sub-surface
# ticket hall were incorporated into an office development,
# which was subsequently rebuilt around 1990. At platform
# level, the station has kept its high brick retaining walls,
# which have been cleaned in recent years.

(The changes of name are mentioned in a separate paragraph.)
--
Mark Brader, Toronto "I seem to have become a signature quote."
-- David Keldsen