That is very close! But it’s cutting one solid out of another not cutting a solid into two “halves”.
And the point is you need to understand the geometry to start with.
Sketchup is just edges and faces and they all stick together, they only separate by making groups or components.
Therefore you need to think about what you are creating.
Yes, you can make the door knob shape you want and select the different parts and group them, but you will be missing faces. So group the shape first and extrude with all faces intact. Simples.
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In case you are still interested this is an example of what has been mentioned in this thread at the most basic level, just using geometry and standard tools.
You can see that your original question about splitting out two pieces from one solid is possible but not without needing to replace faces. Several suggests where given on how, this is just one version.
I see that this thread is still alive.
To complement on my previous proposition, see this SU file as a practice for the use of basic tools.
Missing curved face.skp (214.0 KB)