Cut surface using sphere

Trying to create a surface with spheres cut into it (see attached image). I was able to split the spheres using a surface so that I have my surface with half spheres on one side. However on the other side of the surface it is smooth. I would like to use my initial spheres to cut the surface so that I can see the half sphere. Could someone please offer their expertise? I have searched and searched and can't seem to find instructions on how to cut a surface with a solid. Thank you!