Prepare the Sauce: In a large skillet, melt 1 tablespoon of butter over medium heat. Add the garlic and chopped poblanos, cooking until soft, about 10 minutes.
Add the remaining 2 tablespoons of butter to the pan, and once melted, sprinkle the flour over the garlic and peppers. Stir for a minute to cook the flour into a paste.
Gradually pour in the chicken broth, stirring until the flour dissolves completely. Puree the mixture with an immersion blender until smooth (alternatively, use a food processor or blender).
Bring the sauce to a simmer. Stir in the milk and maintain a low simmer, stirring occasionally, until the sauce thickens. It’s ready when you can swipe a spatula across the bottom of the pan and see the pan for a moment before the sauce covers it again.
Stir in the chopped cilantro and salt, then remove from heat.
Assemble the Enchiladas: Preheat the oven to 375°F. Lightly coat the bottom of a 9x13-inch baking dish with some of the poblano sauce.
Place a spoonful of shredded chicken onto each tortilla, then roll them up and arrange them in the prepared baking dish. Repeat for the remaining tortillas.
Pour the remaining poblano sauce over the enchiladas, then top with the shredded pepper jack cheese.
Bake: Place the dish in the oven and bake for 15-20 minutes, or until the cheese is melted and bubbly.
Serve: Squeeze lime juice over the enchiladas, then garnish with cilantro. Serve hot!