Heat the oven to 450°F. Mix the cumin, ginger, and cinnamon in a small bowl. Reserve 1/4 tsp. of the spice mix in a large bowl. In another large bowl, sprinkle the remaining spices over the lamb and mix with a fork to combine.
Heat a 12-inch skillet over medium-high heat. Add the lamb and all but 1/3 cup of the onion. Season with salt and pepper and cook, stirring and breaking up the meat with the side of a spoon, until the lamb is fully cooked and all of the liquid has evaporated from the pan, about 9 minutes. Stir in 1/2 cup of the chutney and continue to cook for 1 minute. Remove from the heat and keep warm.
Lightly toast the pita breads on a baking sheet in the oven, 6 to 8 minutes, flipping once. Meanwhile, add the remaining 1/3 cup onion and 2 Tbs. chutney, the mango, cilantro, lime juice, and a pinch of salt to the reserved spices to make a salsa; stir to combine. Taste and add more lime juice if it seems sweet (it should be somewhat tangy to offset the sweetness of the meat).
To serve, let diners assemble their own sandwiches by filling the halved pitas with the lamb and salsa.
This recipe also works as an hors d’oeuvre. Cut the pitas into triangles and toss with 4 to 5 Tbs. olive oil and kosher salt. Toast on a baking sheet in a 450°F oven, flipping after 5 minutes, until crisped and slightly browned, 7 minutes total. Top each triangle with some of the lamb mixture and a bit of the salsa.