Prepare the grill for indirect cooking over medium heat (350° to 450°F).
In a large skillet over medium heat, warm the oil. Add the onion and garlic and cook until just starting to turn golden brown, about 3 minutes, stirring occasionally. Add the carrot, celery, and flour and cook for 1 minute, stirring constantly. Add the broth and bring to a boil over medium-high heat. Reduce the heat to medium-low and simmer until slightly thickened, about 10 minutes. Remove from the heat and stir in the turkey, peas, thyme, salt, and pepper.
Put the turkey mixture in a grill-proof 10-inch pie dish and top with the mashed potatoes, spreading them in an even layer. Place the pie dish on a sheet pan to catch any of the juices that might spill out.
Transfer the pie dish on the sheet pan to the grill and place over indirect medium heat. Close the lid and cook until the turkey mixture is bubbling and the potatoes are heated through, 25 to 30 minutes. Remove from the grill and let cool for a few minutes before serving.