Pour ¼ cup of lemon juice into a mixing bowl. Cut the avocados in half, remove the pit, peel them (or scoop them out with a spoon), and cut into ½-inch dice. Immediately, add the avocados to the lemon juice and toss carefully. Next, add the cherry tomatoes and basil and toss again.

Whisk together the remaining 2 tablespoons of lemon juice, ¼ cup olive oil, balsamic vinegar, ½ teaspoon salt, and ½ teaspoon pepper in a small glass measuring cup.

Pour enough vinaigrette over the tomato and avocado mixture to moisten thoroughly. Add the arugula, add more vinaigrette, sprinkle with 1½ teaspoons salt and ½ teaspoon pepper, and toss well. Taste for seasonings (you want the salad well seasoned!) and serve at room temperature.