Preheat the oven to 400°F (200°C). Spray an 8×8-inch (or similar-sized) baking dish with nonstick spray.
Combine the grated white cheddar and grated Monterey Jack; set aside about one-quarter of the total grated cheese for topping and use the remaining cheese in the dip.
In the bowl of an electric mixer, add the softened cream cheese and the portion of shredded cheddar/Monterey Jack reserved for the dip. Beat on medium speed until smooth and creamy, stopping to scrape down the bowl as needed.
Using a large spoon or spatula, fold in the cooked shredded chicken, smoked paprika, cumin, chili powder, onion powder, salt, and pepper until evenly combined.
Stir in the minced garlic, the diced green chiles (with their juices), and the enchilada sauce until the mixture is uniform.
Spread the dip mixture evenly into the prepared baking dish. Sprinkle the reserved grated cheese evenly over the top.
Bake for 25 to 30 minutes, or until the cheese on top is golden and the dip is hot and bubbly in the center.
Remove the dip from the oven and immediately top with the chopped butter lettuce, quartered grape tomatoes, chopped cilantro, and sliced green onions.
Serve immediately with tortilla chips.