Make salsa: In a bowl, combine tomatoes, chiles, and green onions. In a medium saucepan over high heat, bring vinegar to a boil. Add brown sugar and salt and cook, stirring, until dissolved, about 1 minute. Remove from heat. Put ginger, garlic, and dry spices in a bowl. In another medium saucepan, heat olive oil over medium-high heat until rippling. Add ginger mixture and cook, stirring, until fragrant, 1 minute. Remove from heat, stir in seasoned vinegar, and pour over tomato mixture.
Let salsa cool, then cover and chill at least 1 hour and up to 4 hours. About 1 hour before serving, cut avocados into 3/4-in. chunks, add to salsa, and bring to room temperature.
Make chicken: In a large bowl, combine olive oil, lime juice, and seasonings. Turn chicken in mixture to coat. Let stand at room temperature, turning occasionally, 30 to 45 minutes.
Heat a grill to high (450° to 550°). Lift chicken from marinade (discard marinade) and grill chicken, covered, turning often, until no longer pink in center, 15 to 20 minutes.
Transfer chicken to a platter and spoon salsa on top, saving half the liquid for another use (such as salad dressing).
Note: Nutritional analysis is per serving.