1. To prepare carpaccio, combine first 4 ingredients. Peel and slice 3 broccoli stalks into 2 x 1 x 1-inch rectangles, squaring off their sides. Slice broccoli lengthwise into 1/4-inch planks. Combine broccoli planks, 3 tablespoons juice, and 1/4 teaspoon salt. Arrange broccoli planks and avocado slices on 6 plates; sprinkle with basil mixture.
2. To prepare salad, combine olive oil, 1 tablespoon juice, 1/4 teaspoon salt, and pepper in a large bowl, stirring with a whisk. Add 2 cups matchstick-cut broccoli stalks, salad greens, and onion; toss to coat. Divide salad evenly among plates.