(This answer is specific to the context of the question, which is about doing this construction out of Quirk's time dependent gates.)
It's not possible to have proper half-speed time-dependent rotations in Quirk. Behind the scenes there is actually a value $t$ that is varying from 0 to 1 and then cycling. If you managed to find a construction to cut the angle in half, your rotation would jump from 180 degrees to 0 degrees when $t$ went from 1 to 0.
If that's acceptable, then the simplest way to achieve it is by using the parameterized rotation gates and a counting gate:
An alternative trade-off do is define double-speed rotations by using pairs of time-dependent rotation gates, and then work in units of $2t$ instead of $t$. You can use custom circuit gates to fuse them into a single block. If custom gates annoy you (they are by far the weakest part of the UX), the time-dependent gradient gate gives you a series of faster and faster rotations: