As I was watching a tutorial from Daniel Lepik on his Instagram, I noticed a pretty nice trick where he seems to bevel a face after Inset/Extruding inwards, which enables him to make a circular or rounded hole or object inside a square face:
I have redrawn the screenshots here to make it more legible:
It seems like a simple and efficient solution to create round meshes inside faces while limiting the number of polygons and respecting good topology. Also, it doesn't require a modifier which would impact the whole mesh.
However, in Blender, I cannot find any options for Beveling Inset or Extruded faces. I looked in the tool shelf while Extruding / Insetting, nothing. I tried the Bevel shortcut when the face was selected, nothing. Google, nothing.
Can this be done in Blender?
Edit: Following Legoman's answer, I was able to subdivide in a way close to what I am trying to do, but still not sure it is the best way:
Technique 1 ctrl + b + v then to round the shape alt + shift + 1
Technique 2 (sorry about the weird moiré effect at the beginning)