Position a rack in the center of the oven and preheat to 350 degrees . Spread the walnuts on a small baking sheet and bake until toasted, about 10 minutes. Let cool.
Meanwhile, using a food processor fitted with the slicing disk, shred the brussels sprouts (or halve them lengthwise and shred crosswise with a very sharp knife).
In a large skillet, heat the EVOO over medium-high heat. Add the garlic and cook, stirring occasionally, until golden, about 2 minutes. Add the brussels sprouts, 3/4 tsp. salt and 1/4 tsp. pepper and cook, tossing well, until crisp-tender, about 5 minutes. Stir in the toasted walnuts and spoon into a serving bowl.