Trim the bottoms from the Brussels sprouts and slice them thinly using the slicing blade of a food processor, a sharp knife and cutting board, or a mandoline. Place them on a cutting board and give them a rough chop with a sharp knife.
Place the Brussels sprouts, apples, Pecorino Romano, and dried fruit in a large bowl.
In a small bowl or container, combine the olive oil, lemon juice, shallot, salt, and pepper.
Pour the dressing over the salad and toss to combine well. Transfer to a serving bowl.
Notes
To slice the apple into small thin slices, just cut two flat almost-halves from the apple, just on either side of the core. Place the halves on a cutting board, slice them in half lengthwise, then cut crosswise into thin slivers. Cut the two remaining side chunks from the apple core and slice them crosswise thinly as well. You can also just cut the apple into quarters, core them, and then slice them crosswise.
If you want to use grated Pecorino Romano, that's fine, but adding cheese that has been shaved super thin offers a really nice texture and generous bites of Pecorino. Simply peel off thin slivers of the cheese with a vegetable peeler. Don't worry about the pieces breaking or being of similar size. You can also purchase shaved Pecorino or Parmesan (a good substitution) in many grocery or cheese stores.