Make the turkey: Combine the brown sugar, thyme, coriander, ground ginger, paprika, allspice, cayenne, garlic and 2 teaspoons salt in a small bowl. Pat the turkey dry. Rub the vegetable oil all over the turkey, then rub with the spice mixture. Let sit at room temperature 30 minutes before grilling.
Preheat a grill to medium high and prepare for indirect grilling: On a charcoal grill, bank the coals to one side; on a gas grill, turn off half the burners. Brush the grill grates with vegetable oil. Lay the turkey on the grates skin-side up with the legs closest to the direct-heat side (the dark meat needs more heat). Cover the grill and cook until the turkey is deep golden brown and a thermometer inserted into the thickest part of the thigh registers 165 degrees F, 2 1/2 to 3 hours. Transfer to a cutting board and let rest 15 minutes.
Meanwhile, make the chutney: Combine the plums, shallot, dried apricots, brown sugar, vinegar, fresh ginger, thyme, mustard seeds, allspice, coriander, cayenne and 1 teaspoon salt in a medium saucepan. Bring to a simmer over medium heat and cook until the plums are tender and the juices thicken, 10 to 15 minutes. Remove from the heat and let cool. Serve the turkey with the chutney.
If you're making the turkey on a charcoal grill, you'll need 4 to 5 pounds of charcoal. Add several new briquettes to the coals every 30 minutes to maintain the heat.