For the Poached Chicken: In a medium pot set over medium heat, toast cumin seeds, stirring constantly, until fragrant, about 40 seconds. Add chicken, chicken broth, chiles, and salt. Increase heat to medium-high and cook until broth is just shy of a simmer and the temperature registers between 150º and 160ºF (65 to 70ºC) on an instant-read thermometer, about 12 minutes. (See notes.)

Reduce heat to low to maintain the temperature of the broth. (To help maintain the temperature, increase or reduce the heat as needed, and/or re-position the pot over stovetop burner so that the heat source only partially touches bottom of the pot.) Cook until an instant-read thermometer inserted into thickest portion of chicken registers 150°F (65ºC), 30 to 35 minutes. Carefully transfer chicken to a large bowl, and let rest until chicken is cool enough to handle, about 10 minutes. Using 2 forks, shred chicken; set aside. Using a fine-mesh strainer set over a large bowl, strain broth; discard solids. Set aside 2 ½ cups (592ml) broth. Reserve remaining broth for another use.

While chicken cooks, prepare the casserole: Adjust oven racks to top third and lower third positions and preheat oven to 400ºF (205ºC). Arrange tortillas on 2 large rimmed baking sheets (they will be slightly overlapping). Lightly spray both sides with cooking spray. (Alternatively, use pastry brush to lightly brush tortillas with a thin layer of neutral oil.)

Bake until slightly crisp and browned in spots, about 14 minutes, flipping tortillas halfway through and rotating baking sheets between top and bottom racks. Remove from oven and let cool until ready to use. (Tortillas will continue to crisp as they cool.)

In a large pot or Dutch oven set over medium-high, melt butter. Add onion, bell pepper, poblano, and garlic; cook, stirring occasionally, until softened, about 8 minutes. Add flour, chili powder, and granulated chicken bouillon. Cook, stirring often, until fragrant and flour just begins to stick to bottom of the pot, about 1 minute. Add diced tomatoes and green chiles, cream, and 2 ½ cups reserved broth; bring to a simmer over medium-high. Simmer, stirring occasionally, until slightly thickened, about 4 minutes. Remove from heat, and stir in cilantro, sour cream, salt, and shredded chicken until thoroughly combined.

To assemble, place 6 tortillas in the bottom of a 9- by 13-inch baking dish. Spoon 3 cups of the chicken mixture over tortillas, and top with ½ cup of Monterey Jack cheese. Repeat with second layer. Top with remaining 6 tortillas, remaining 2 cups chicken mixture, and remaining 1 ½ cups cheese.

Bake, uncovered, at 400°F on top third rack of oven until bubbling and browned, 15 to 20 minutes. Garnish with additional cilantro.