I tweaked a little bit based on the understanding of Elias's fragmented sphere example. I wanted the extruded fragments triggered by mouse to be a enclosed shape. Therefore, I need to extent the vertex which shared with the intersected face as well.

Code hosted in the codeSandBox.

So the result looks like this:

However, it didn't cover the indices that didn't share with them but still were part of the side faces.

Also, I wanted to detect the neighborhood faces around the intersected face but didn't know how. Any suggestions?

I finally got to know how the Voronoi diagram worked after looking into its theory. Here's the code and the demo. The art of code explained the code very clearly!