Prepare 3 cups diced cooked chicken by shredding or dicing rotisserie chicken or poaching two large chicken breasts in simmering water for 10–15 minutes until cooked through, then dice.
Make the dressing: in a small bowl whisk together olive oil, mayonnaise, 1 tablespoon fresh lime juice, 1/4 to 1/2 teaspoon ground cumin, and salt to taste; stir in Green Tabasco Sauce starting with 1 teaspoon and add a second teaspoon if desired.
Place the diced chicken in a large bowl and add about half the dressing; toss to coat and let the chicken marinate while you prepare the vegetables.
Trim and wash the radishes, then cut them in halves or quarters lengthwise and slice so each piece shows the red edge; add to the bowl with the chicken.
Slice the green onions (white and green parts) and add them to the chicken and radishes.
Add the remaining dressing to the chicken mixture (or as much as you prefer) and stir to combine.
Dice the avocados into 1-inch pieces, place them in a small bowl, and toss with 1 tablespoon fresh lime juice to prevent browning.
Gently fold the dressed avocado into the chicken mixture, season to taste with fresh-ground black pepper and additional salt if needed, then serve.