Likewise see if you can imagine squishing and stretching a putty coffee mug or a straw into a torus. They both have 1 hole. Like this: https://i.ytimg.com/vi/9NlqYr6-TpA/maxresdefault.jpg

If you want to consider a more restrictive collection of modifications, like, say you want to preserve angles or something inside the shape, then that would be a different question. Uh, I guess if you had a surface which was generally smooth but possibly has some boundaries and/or corner edge things, and/or corner point things (e.g. sticking two cones together), you could also ask about how many cuts combined with conformal maps [edit : I think conformal maps isn't the thing I'm looking for. Yeah, I think I'm looking for something related to developable surfaces maybe] would be needed to flatten such a thing out? (so this could maybe produce a similar question for which the one or half or whatever cut between the edge of the cone and the center, could be an answer?)

From the topological standpoint, you do not need to cut the sock for this - it is already “the same as” (homeomorphic to) a piece of the plane surface.