Preheat oven to 350 degrees F. Toss walnut halves with walnut oil (or leave plain) and spread out on a cookie sheet pan or pie plate. Toast in oven for 5 - 7 minutes, until they begin to smell nutty and turn golden brown. Remove from oven, transfer nuts from hot baking pan to plate, and let cool.
Thinly slice red onion on mandoline, set aside.
Quarter the cabbage and slice very thinly using a mandoline or chef's knife. (I find it easiest to leave the core in when I'm using a mandoline. But if I'm chopping with a chef's knife, I remove the core from each quarter, then slice.)
Slice apple into matchsticks and set aside. Here's how I like to do it: Stand apple upright and slice off each side leaving the core. Slice off the remaining two sides. Thinly slice each piece into planks, flip on their sides and slice into matchsticks.
Put oil and balsamic vinegar into a large (12 - 14 inch diameter) sautée pan and heat over medium-high heat. Add the onion and cook for 30 seconds. Add the cabbage and stir with a pair of tongs for a couple minutes, until just wilted. Season with sea salt and freshly ground black pepper then taste—add more vinegar, if necessary, to sharpen the flavors. Add the apple slices, goat cheese, and walnuts. Toss gently before serving.