We derived and for a two dimensional continuous object rotating in an axis perpendicular to its plane, but we can apply it to non-pancake-like objects with a few caveats.
First consider some symmetric object that's rotating about a symmetry axis in zero gravity:
there is no torque on the bearing.
Now take off one of the arms
Now the object will not continue to spin around. It is ``off balance". To insure it continues to rotate around as it did before you need to change the construction of the bearing:
Now the bearing provides a torque to hold the object at the right angle. If we apply an external torque to the red ball, it will cause a torque to be applied to the bearing. The bearing torque keeps the objects rotating about in the horizontal direction. In this case we can still write , but keep in mind that additional torques are being generated to keep the object rotating around like we want it to.
Now what happens to ? In the first picture, the angular momentum vector points along the axis of rotation. In the second it does not. Our derivation of assumes the former case. We have to be careful to only use this formula when rotation is along an axis of symmetry.