In a resealable plastic bag, combine 3 tablespoons olive oil, 3 tablespoons lime juice, the smashed garlic and 1 1/2 teaspoons salt. Add the chicken and let marinate at room temperature for 15 minutes. In a grill pan or skillet, cook the chicken over medium-high heat until cooked through, 6 to 8 minutes; let cool slightly and shred. Set aside.
In a medium saucepan, heat the remaining 1 tablespoon oil over medium heat, add the rice and cook, stirring, until golden, about 5 minutes. Stir in the chicken broth, tomato sauce, cumin, 1 teaspoon salt and the chopped garlic. Cover and bring to a boil, then simmer until the liquid is absorbed, about 20 minutes.
Preheat the oven to 350 degrees . Place each tortilla on a sheet of foil and top with the rice, beans, reserved chicken and cheese. Wrap the burritos snugly in the foil and heat for 10 to 15 minutes before serving.
Meanwhile, in a medium, nonreactive bowl, combine the mango, chiles, cilantro and remaining 1 tablespoon lime juice; let sit for 15 minutes (or refrigerate overnight). Serve the salsa over the chicken burritos.