Preheat oven to 400°F.
Place sweet potatoes and water in a microwave-safe bowl. Cover with plastic wrap; pierce a few holes in the top with a fork. Microwave on High for 4 minutes.
Meanwhile, heat oil in a large skillet over medium-high heat. Add chicken and cook until browned, 4 to 5 minutes per side. Transfer the chicken to a clean cutting board and let stand for 5 minutes. Cut into 1-inch strips.
Add poblanos, shallots and garlic to the pan and cook over medium-high heat, stirring occasionally, until the shallots are lightly browned, about 2 minutes. Add broth, quinoa, wine, salt, cumin, cinnamon and cayenne. Bring to a boil. Remove from heat and stir in the sweet potatoes and chicken.
Spoon the mixture into a 9-by-13-inch broiler-safe baking dish. Cover with foil. Bake until the sweet potatoes and quinoa are tender and the liquid is fully absorbed, 35 to 40 minutes.
Remove from oven; increase oven temperature to broil. Uncover the casserole and sprinkle with cheese. Broil 8 inches from the heat source until golden brown, about 5 minutes. Sprinkle with cilantro. Let cool for 5 minutes before serving.
