In 10-inch nonstick skillet, heat oil over medium-high heat until hot. Add garlic and onion; cook about 5 minutes, stirring constantly, until evenly roasted.
Remove husks and rinse tomatillos well. Core top stem area of each; cut tomatillos into quarters. In large food processor, place roasted garlic and onion, tomatillos, jalapeño chiles, cilantro, 1 teaspoon of the salt and the sugar. Cover; process about 15 seconds or until almost smooth.
Pour 2 cups mixture into small serving bowl; cover and refrigerate to use as salsa. To make marinade, add remaining 1 teaspoon salt and the cumin to remaining mixture in food processor. Cover; process 10 seconds.
Place 6 chicken breasts in each of 2 gallon-size resealable freezer plastic bags; divide marinade between bags. Seal bags; shake to evenly distribute marinade and coat chicken. Refrigerate at least 8 hours but no longer than 24 hours; turn bags over at least once while marinating.
Take salsa, bags of chicken, tongs, metal spatula and a serving platter to potluck. Salsa should be room temperature when served with grilled chicken.
Heat gas or charcoal grill. Carefully brush oil on grill rack. Place chicken on grill over medium heat; discard marinade. Cover grill; cook 12 to 15 minutes or until juice of chicken is clear when center of thickest part is cut (170°F), turning once. Serve each chicken breast with about 2 tablespoons salsa.