Pour balsamic vinegar into a small jar with a lid; add garlic, the salt and pepper, and 3 tablespoons of the oil. Tightly seal jar; shake well. Set aside. In a small saucepan, bring 1/2-inch salted water to boil; add beans. Cook until tender-crisp, about 5 minutes; cover pan when water returns to boil. Drain; transfer to a medium bowl. Spoon 3 tablespoons reserved dressing over hot beans; toss. Cool beans to room temperature. Meanwhile, in same saucepan, bring 1/2-inch salted water to boil; add beets. Cook until tender when pierced with knife, about 8 minutes; cover pan when water returns to boil. Drain; transfer to a small bowl. Spoon remaining dressing over hot beets; toss. Cool beets to room temperature. Just before serving, gently toss together reserved beans with fennel, onion, and avocado; add remaining tablespoon of the oil, if desired. Adjust salt and pepper to taste. Arrange mixture on 6 to 8 salad plates; top with reserved beets and cheese. Garnish with parsley, if desired. (Serve salad at room temperature; refrigerate leftovers.)
Makes 6 to 8 servings.