In a small bowl, stir together the red onion, ¼ cup lime juice, and a sprinkle of salt.
In a medium bowl, combine the garlic with ginger, cumin, turmeric, garam masala, 1 teaspoon of salt, and yogurt. Set aside ½ cup of the yogurt mixture in a small bowl. Cover the rest and refrigerate.
Add chicken to the ½ cup of yogurt mixture. Toss to coat; set aside.
In a medium bowl, combine the dates, tamarind, ginger, cumin, and a pinch each of cayenne and salt. Pour ⅓ cup boiling water over the date mixture. Mash with a fork, until it forms a chunky paste. Set aside.
Heat a large skillet over medium-high heat. Working in batches, sear chicken until golden brown and cooked through, 5 to 6 minutes. Transfer to a serving dish.
To serve, spoon rice and chicken into lettuce cup. Top with pickled onion and cilantro. Serve with chutney and reserved yogurt sauce for dipping.