Prepare the Sauce:
In a large pan, melt 1 tablespoon of butter over medium heat. Add the smashed garlic and chopped poblano peppers. Cook for about 10 minutes until softened.
Add the remaining 2 tablespoons of butter to the pan. Once melted, sprinkle the flour over the garlic and peppers. Stir and cook for about 1 minute, forming a thick paste that coats the veggies.
Gradually pour in the chicken broth while stirring to avoid clumps of flour. Using an immersion blender, puree the mixture until smooth.
Bring the sauce to a simmer, then add the whole milk, stirring frequently. Keep the sauce at a low simmer until it thickens (about 5 minutes). Once thickened, stir in the chopped cilantro and salt. Turn off the heat.
Preheat the Oven:
Preheat your oven to 375°F (190°C).
Assemble the Enchiladas:
Coat the bottom of a 9×13-inch baking dish with a light layer of the poblano sauce.
Place a portion of the shredded rotisserie chicken onto each warmed tortilla. Roll them up tightly and place them seam-side down in the baking dish.
Once all tortillas are filled and placed in the dish, pour the remaining poblano sauce evenly over the top of the enchiladas.
Add Cheese & Bake:
Sprinkle the shredded pepper jack cheese evenly over the top of the enchiladas.
Bake, uncovered, in the preheated oven for 15-20 minutes, or until the cheese is melted and the sauce is bubbling.
Garnish & Serve:
After baking, squeeze fresh lime juice over the enchiladas and garnish with additional chopped cilantro. Serve warm and enjoy!