Finding a word ladder using Breadth First Search
If you worry about the performance, the first thing to try is to hook up the profiler. Besides that, constructing another Ladder at each level seems wasteful.


is_changeable is named misleadingly; my first impression was that you are testing a mutability of something. It turned out that the method test for a Hamming distance to be 1. I recommend to name it hamming_distance, make it return int (or size_t or whatever you find appropriate) ant let the client test the result for 1.
Use of lambdas borders to abuse. If you feel a need to comment what lambda is doing, it is a strong indication that the lambda wants to have a name, i.e. be a normal function.