Trim the stem end of the Brussels sprouts and remove any discolored or damaged outer leaves. Cut each sprout in half from top to bottom. Shred Brussels sprouts with a sharp knife or a food processor fitted with a slicing disc. Remove the ribs from the kale, and discard. Chop remaining leaves into bite-sized strips. Place sprouts and kale in a large bowl, and set aside.
Add lemon juice, anchovy paste, garlic, red pepper and kosher salt to a small bowl, whisking to combine. Whisk in olive oil until full incorporated. Drizzle over the prepared greens, and using your hands, massage the dressing into the shredded and chopped leaves. Toss in pecorino and toasted walnuts. Let the salad sit at room temperature for 20 - 25 minutes until the leaves are tender. Season to taste with additional kosher salt and black pepper.
To serve, spoon salad in a big bowl, and top with a sprinkling of pecorino cheese. Store any leftover salad covered in the refrigerator for up to 1 day.
* When grating a large amount of hard cheese, I like to use a food processor fitted with a steel blade. Simply chop pecorino up into large chunks, and toss into the food processor. Blitz until the texture is finely ground.