Trim the stem end of each Brussels sprout. Cut each sprout in half lengthwise. Now cut each half Brussels sprout into very thin wedges and set in a bowl.
Place a large skillet over medium, to medium-high heat. Add the walnuts and cook until lightly toasted, about 2 minutes. Scoop walnuts out with a slotted spoon and set in a small bowl.
Add the Brussels sprouts to the skillet and cook and stir 2 to 3 minutes. Add the apple juice and bring to a simmer. Simmer 2 minutes, until sprouts soften slightly and turn a bright green. Mix the cubed apple, walnuts, salt and pepper into the salad. Transfer salad to a shallow serving bowl. Drizzle with the 1 Tbsp. olive oil and vinegar, and then serve.