I am having trouble figuring out what Thoreau meant by "to cut a broad swath and shave close."
Getting meaning out of just that quote is possible, but it makes more sense when taken with the full sentence it came from.
"I wanted to live deep and suck out all the marrow of life, to live so sturdily and Spartan-like as to put to route all that was not life, to cut a broad swath and shave close, to drive life into a corner, and reduce it to its lowest terms, and, if it proved to be mean, why then to get the whole and genuine meanness of it, and publish its meanness to the world; or if it were sublime, to know it by experience, and be able to give a true account of it in my next excursion."
What Thoreau is saying with this sentence is that he wants to "live life to the fullest." An English teacher probably would now talk about "carpe diem" and maybe even show a scene from Dead Poets Society. Carpe diem is the original way to express the now current "you only live once (YOLO)" feeling.
The big difference for Thoreau though is that in order for him to do this, he had to get rid of anything that he felt was distracting. So out goes society and here comes living in tiny shack on Walden Pond for 18 months or so. He wants to experience life at its simplest and really get down to living. Thoreau wants to cut away a lot of what he feels is getting in the way of living close to nature (cut a broad swath). He wants to get close to the true essence of life and living (shave close).