Preheat the oven to 375°F (190°C).
Place the bone-in chicken breasts in a covered roasting pan or deep casserole and add 2 cups water; cover and roast in the oven for about 1 hour, or until the chicken reaches 165°F internally. Reserve the cooking liquid (broth) for the sauce and let the chicken cool slightly.
Discard the skin, remove the meat from the bones, and shred the chicken with two forks. In a mixing bowl, combine the shredded chicken with 3/4 cup of the shredded cheddar, chili powder, cumin, salt, and black pepper.
Spoon an even portion of the chicken mixture onto each tortilla, roll tightly, and place seam-side down in a 9 x 13-inch casserole pan.
Make the green chile sauce: melt the butter in a medium saucepan over medium-low heat. Whisk in the flour and cook for 1 minute until smooth. Gradually whisk in 2 cups reserved chicken broth, increase heat to medium, and cook, whisking, until the sauce thickens to a gravy consistency.
Remove the sauce from heat and stir in the Greek yogurt and diced green chiles until smooth and combined.
Pour the green chile sauce evenly over the arranged enchiladas and sprinkle the remaining 1 cup shredded cheddar on top.
Cover the casserole with foil and bake at 375°F for 20 minutes. Remove the foil and bake an additional 5–10 minutes, until the cheese is melted and the casserole is heated through.
Let the enchiladas rest a few minutes before serving. Garnish as desired.