Preheat oven to 375 degrees. Grease 9x13 pan and set aside.
Add 1 tablespoon of butter to a medium skillet. Once melted, add diced peppers. Cook until soft, about 8 minutes. Add garlic and cook until fragrant. Add 2 more tablespoons of butter. Once melted, sprinkle flour over the top and then whisk together until a paste forms. Cook for 1 minute.
Add chicken stock about ¼ cup at a time to thin out the paste but it still stays thick-ish. Reduce temperature to low. Using an immersion blender, blend the mixture until almost no pepper chunks are seen and the mixture is green. Turn temperature back up to medium.
Once the mixture is simmering, slowly add milk to the mixture, about ⅓ cup at a time. Bring to a simmer and thicken. Turn off heat and add cilantro, salt, and lime juice. Stir together.
Grab greased pan and spoon about ½ cup of sauce into the pan and spread evenly over the bottom. Heat flour tortillas by wrapping in a damp paper towel and microwaving for 30 seconds. Flip tortillas and microwave another 30 seconds.
Place about 3 tablespoons of shredded chicken along the center of the tortilla. Roll up into a cylinder with both ends open. Place seam-side down in the pan. Repeat with the rest of the tortillas. Cover enchiladas evenly with sauce and then with shredded mozzarella.
Bake in preheated oven until cheese starts to turn golden brown, about 25 minutes. Remove from oven and let cool for 10 minutes before serving.