Lightly coat a 6-quart slow cooker with cooking spray. Arrange half the chiles in an even layer in the slow cooker. Sprinkle evenly with ½ cup each Cheddar and pepper Jack cheese. Top evenly with chicken. Top evenly with remaining chiles; sprinkle with ¼ cup each pepper Jack and Cheddar.
Whisk evaporated milk, eggs, cornstarch, salt, garlic powder, pepper, and paprika together in a large bowl until well combined.
Pour egg mixture evenly over chile mixture in cooker.
Cover and cook on Low until set around the edges and just set in the center, 3 ½ to 4 hours.
Remove lid. Dab top of casserole gently with paper towels to absorb excess moisture, if needed. Top with enchilada sauce, remaining ½ cup Cheddar, and remaining ¼ cup pepper Jack cheese.
Continue to cook, uncovered on Low, until cheese has melted, 15 to 20 minutes. Turn cooker off.
Garnish with cilantro. Serve with sour cream.
