Preheat broiler with a rack set about 4 in. from heat. Set tomatoes, onion, and jalapeño on a foil-lined rimmed baking sheet. Broil, turning as needed, until blackened, 15 to 25 minutes; as done, transfer to a bowl and let cool.
Peel, stem, and seed chile. In a food processor, pulse vegetables (including blackened parts from tomatoes and onion) and their juices until chopped. Add remaining salsa ingredients and pulse until finely chopped. Transfer to a bowl.
Cut chicken into 1- to 1 1/2-in. chunks. In a bowl, combine 2 tbsp. oil, the cumin, salt, chopped cilantro, and chicken.
Heat a 12-in. frying pan (not nonstick) over high heat. Add remaining 1 tbsp. oil; swirl to coat. Cook chicken, stirring and turning once, until lightly browned but still moist, 5 minutes.
Stir cilantro leaves into chicken; transfer with a slotted spoon to a platter. Serve with salsa and skewers.
*Toast over medium heat until fragrant, 3 minutes.
Make ahead: Through step 4, up to 1 day, salsa and chicken chilled separately. Microwave salsa until room temperature (about 3 minutes) and chicken, stirring occasionally, until hot (about 4 minutes).