How to connect multi-strand sheathed wire?
Sounds like a bad plan. 

"We want to cut the current cable in halves, and connect a new section of cable in between." 

And then what? Hide the splices in a wall or under cabinets? No. Leave it on the floor or stapled to the wall? No. Run a completely new wire into the stove and get a replacement plug? Yes.

Or do it the right way and install an outlet where the stove now goes.