Combine the curry powder, allspice, chopped thyme, 1 teaspoon salt and a few grinds of pepper in a medium bowl. Add the chicken and toss to coat; set aside.
Skim 2 tablespoons of the solid cream off the top of the coconut milk; transfer to a large Dutch oven or pot and melt over medium-high heat. Add the chicken and cook, turning, until browned, about 5 minutes; transfer to a plate.
Add the onion, garlic, cilantro, chile and thyme sprigs to the pot and cook, stirring, until the onion is softened, about 5 minutes. Return the chicken to the pot along with 1/4 cup water; stir. Add the coconut milk (including any remaining coconut cream from the can). Tear 1 lavash into small pieces; add to the pot and bring to a simmer. Reduce the heat to medium low, cover and cook until thickened, about 15 minutes.
Just before serving, warm the remaining lavash in the microwave. Remove the thyme sprigs from the pot and stir in the peas and carrots. Cook until warmed through, about 2 minutes. Season with salt and pepper. Top with cilantro and serve with the remaining lavash.