In a large pan, melt 1 tablespoon of butter over medium heat. Add the smashed garlic and chopped poblano peppers to the pan, cooking for about 10 minutes until they soften and become aromatic. After they are tender, stir in 2 more tablespoons of butter, followed by sprinkling the flour over the mixture. Stir continuously for 1 minute until a thick paste forms.

Gradually pour in 1 1/4 cups of chicken broth, stirring until the mixture is smooth and combined. Use an immersion blender to puree the sauce until silky. Bring it to a gentle simmer, then add 1 1/4 cups of whole milk. Stir continuously for 5-7 minutes until the sauce thickens slightly, and then mix in the chopped cilantro and salt before removing from heat.

Preheat your oven to 375°F (190°C). In a 9x13 baking dish, pour a thin layer of the creamy poblano sauce on the bottom to prevent sticking. Take each warmed corn tortilla and fill it with shredded rotisserie chicken, then roll it tightly and place seam-side down in the dish. Continue this process until all tortillas are filled and snugly arranged in the baking dish.

Once all the enchiladas are in the dish, pour the remaining creamy poblano sauce evenly over the top. Sprinkle the shredded pepper jack cheese generously over the sauce, giving a rich, creamy finish. Bake uncovered in the preheated oven for 15-20 minutes, until the cheese is melted and the sauce is bubbly, creating an inviting aroma in your kitchen.

After removing the baking dish from the oven, squeeze fresh lime juice over the enchiladas to brighten the flavors. Garnish with a sprinkle of chopped cilantro for a fresh pop of color. It's now time to enjoy these delectable creamy poblano chicken enchiladas with your loved ones, savoring every comforting bite!