For the Chicken and Sauce
1 (3 1⁄2 lb.) chicken, with the neck, but giblets removed
1 small carrot (3 oz.)
1 large white onion (10 oz.)
3 unpeeled garlic cloves
1⁄2 tsp. cumin seed
2 tbsp. unsalted butter
2 cups diced white onion
4 garlic cloves, minced
2 tbsp. all-purpose flour
3 cups roasted, peeled and chopped Hatch green chiles (from about 30 if using fresh chiles)
6 cups hot chicken broth (see below)
Salt and pepper
1⁄4 tsp. dried oregano
For the Enchiladas and Garnish
12 soft corn tortillas
2 1⁄2 cups (7 oz.) grated Oaxacan or Monterey Jack cheese, or a mix
2 cups (10 oz.) crumbled queso fresco
4 scallions, thinly sliced (3⁄4 cup)
1 cup coarsely chopped cilantro leaves and tender stems (from about 1 bunch)
Crema, crème fraîche, or sour cream
Thinly slivered crisp lettuce, tomato slices and radishes (optional)
Make the chicken: In a large pot or Dutch oven, add the chicken, chicken neck and enough water just to cover. Add the carrot, onion, and garlic cloves and bring to a simmer. Let simmer until the meat is cooked through and the broth is flavorful, about 40 minutes.
Strain, reserving both the broth and the chicken. (If desired, you can continue to simmer the chicken neck and vegetables in the stock to further concentrate the flavor while the chicken cools, another 30 minutes.) Let the chicken rest until cool enough to handle. Shred the meat and discard the bones and skin. You should have 4 cups meat. Set aside. Reserve 6 cups of broth for the sauce, plus 2 cups more for softening tortillas.
Make the sauce: In a small, dry skillet over medium-high heat, add the cumin. Cook, stirring or shaking the pan occasionally, until fragrant, about 3 minutes. Remove, then finely grind.
In a large saucepan over medium-high heat, melt the butter. Once hot, add the onions and cook, stirring occasionally, until softened and barely colored, about 8 minutes. Stir in the minced garlic and let cook 1 minute, then sprinkle the flour into the pan and stir to incorporate. Add the chiles. Season generously with salt and lightly with black pepper, then add the cumin and oregano; stir well. Gradually whisk in the chicken broth and bring to a simmer. Simmer for 10 minutes (sauce should thicken slightly). Taste and adjust the seasoning as needed, then set aside. (Sauce and chicken can be prepared up to 1 day ahead. Cover and refrigerate.)
Make the enchiladas: Preheat the oven to 400°. Ladle 1 1⁄2 cups of the sauce into a 9-by-13-inch glass baking dish. Heat the remaining chicken broth (alternatively, prepare a steamer). To soften the tortillas so they can be rolled without breaking, dip them very briefly in the hot broth, or steam them, until softened slightly. (Traditionally, tortillas are lightly fried to soften.)
Working one at a time, distribute about 3 tablespoons of the shredded chicken down the center of each tortilla. Roll the tortilla loosely into a cigar shape to cover the filling, then transfer seam side down to the prepared baking dish. Repeat with the remaining tortillas and chicken, placing the enchiladas snugly side by side. Ladle the remaining sauce over the top to coat generously. Sprinkle the cheese atop the sauce, cover with foil and bake until the enchiladas are heated through and the sauce is beginning to bubble, about 20 minutes. Remove the foil and bake until the top is browned and bubbling, 15-20 minutes more.
Remove the enchiladas and sprinkle with the queso fresco, scallions, and cilantro, Divide among 6 individual plates and drizzle or dollop each portion generously with crema or sour cream. Garnish each with the lettuce, tomato and radish, if using. Serve immediately.