Pre-heat oven to 450º.
Clean and trim ends and loose leaves off of brussels sprouts and add to a bowl. Drizzle 1 - 2 tablespoons over them and mix well. Sprinkle salt and pepper to taste. Add to a baking dish and bake for 20 minutes, stirring twice.
Meanwhile, toast walnuts in a saute pan over medium heat.
Make vinaigrette by mixing lemon juice, 1/4 cup of olive oil, and 1/2 teaspoon of salt.
When brussels sprouts are done, transfer to a bowl and add walnuts and pomegranates. Mix well and then drizzle vinaigrette over top and mix again.
Plate and add sliced or shredded cheese.