Preheat a gas grill to high. Brush chicken with 1 Tbsp. oil. Season with salt and pepper.
Whisk together lime juice, sugar and ginger in a large mixing bowl until sugar has dissolved fully. Whisk in remaining 3 Tbsp. oil until well combined. Add avocado, radishes, scallions and cilantro to bowl. Season with salt; mix until completely coated in lime-juice dressing.
Turn grill to medium-high and oil grates. Grill chicken with cover closed, turning once, until cooked through (no pink remains), about 10 minutes total. Transfer chicken to a cutting board and loosely cover with foil. Let rest for about 5 minutes. Thinly slice chicken and add to bowl with avocado mixture. Toss together until combined. Divide filling equally among tortillas and serve immediately.