1 poundmedium beefsteak-type tomatoes, quartered, seeds squeezed out
2 serrano chiles, thinly sliced
1/2 cupthinly sliced green onions
1/2 cupdistilled white vinegar
2 1/2 tablespoonspacked light or dark brown sugar
1 1/2 teaspoonskosher salt
4 teaspoonsminced fresh ginger
1 tablespoonminced garlic
2 teaspoonsmustard seeds
2 teaspoonsfreshly cracked black pepper
2 teaspoonsground cumin
1/2 cupextra-virgin olive oil
2 firm-ripe avocados, cut into 3/4-in. chunks
1/4 cupextra-virgin olive oil
1/4 cupfresh lime juice
1/2 cupchopped cilantro
1 tablespoonground cumin
1/2 teaspoonkosher salt
1/2 teaspoonfreshly ground black pepper
4 chicken breast halves with bone and skin (2 1/2 lbs. total)
1. Make salsa: In a bowl, combine tomatoes, chiles, and green onions. In a medium saucepan over high heat, bring vinegar to a boil. Add 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.
2. Let salsa cool, then cover and chill at least 1 hour and up to 4 hours. About 1 hour before serving, stir avocados into salsa and bring to room temperature.
3. Make chicken: In a large bowl, combine olive oil, lime juice, cilantro, cumin, salt, and pepper. Turn chicken in mixture to coat. Let stand at room temperature, turning occasionally, for 30 to 45 minutes.
4. Prepare a grill for high heat (450° to 550°; you can hold your hand 5 in. above cooking grate only 2 to 4 seconds). Lift chicken from marinade (discard marinade) and grill chicken (cover gas grill), turning often to prevent scorching, until no longer pink in center, 15 to 20 minutes.
5. Transfer chicken to a platter and spoon salsa on top, saving half of liquid for another use (such as salad dressing).