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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 1/2 cup each Cheddar and pepper Jack cheese. Top evenly with chicken. Top evenly with remaining chiles; sprinkle with 1/4 cup each pepper Jack and Cheddar.
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Whisk evaporated milk, eggs, cornstarch, salt, garlic powder, pepper, and paprika together in a large bowl until well combined.
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Pour egg mixture evenly over chile mixture in cooker.
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Cover and cook on Low until set around the edges and just set in the center, 3 1/2 to 4 hours.
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Remove lid. Dab top of casserole gently with paper towels to absorb excess moisture, if needed. Top with enchilada sauce, remaining 1/2 cup Cheddar, and remaining 1/4 cup pepper Jack cheese.
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Continue to cook, uncovered on Low, until cheese has melted, 15 to 20 minutes. Turn cooker off.
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Garnish with cilantro. Serve with sour cream.