Stir together garlic and ginger in a large bowl. Transfer half of the garlic mixture (about 1 ½ tablespoons) to a small bowl; set aside. Add chicken and turmeric to large bowl with remaining garlic mixture; toss to coat. Cover and chill for 30 minutes.
Heat oil in a large Dutch oven over medium. Add onion and Thai chile; cook, stirring often, until onion is softened and lightly browned, about 15 minutes. Add reserved half of the garlic-ginger mixture; cook, stirring constantly, until fragrant, about 1 minute. Stir in coriander and chile powder; cook, stirring constantly, until evenly combined, about 20 seconds.
Add chicken to Dutch oven; cook, turning occasionally, until browned on all sides, 6 to 8 minutes. Stir in 1 cup water and salt. Bring to a simmer over medium. Reduce heat to low; cover and simmer, stirring occasionally, until chicken is mostly cooked through, 10 to 15 minutes.
Stir chopped mango into Dutch oven; cover and cook, stirring occasionally, until a thermometer inserted into thickest portion of chicken registers at least 165°F and mango is softened, 5 to 8 minutes. Stir in mango chutney and garam masala. Cook, uncovered, until liquid reduces slightly, 2 to 3 minutes. Remove from heat. Drizzle with lime juice. Garnish with cilantro; serve with parathas or rice.
