Soak sliced shallots in a cup of ice water for 10 minutes. Stir frequently, drain and pat dry. Set aside.
In a non-reactive bowl, whisk the red wine vinegar, garlic, mayonnaise with salt and pepper until smooth and well incorporated. Continue to whisk steadily while drizzling in the extra virgin olive oil until thoroughly incorporated into the dressing. Place avocados in a separate non-reactive bowl and top with 3 tablespoons dressing, toss until evenly coated. Set aside.
In another nonreactive bowl, toss prepared shallot, tomatoes, basil and remaining dressing until evenly coated. Add salt and pepper to the tomato mixture to taste. Set aside.