Preheat the oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Melt 1 tablespoon (14 g) of butter in a large skillet over medium heat. Add the smashed garlic and chopped poblano peppers. Cook for about 10 minutes until the peppers soften.
Add the remaining 2 tablespoons (28 g) of butter to the skillet. Once melted, sprinkle the flour over the vegetables. Stir continuously for 1 minute to form a paste.
Gradually whisk in the chicken broth to dissolve any flour clumps. Use an immersion blender to puree the mixture in the pan until smooth.
Bring the pureed mixture to a gentle simmer. Stir in the whole milk. Cook at a low simmer, stirring often, until the sauce thickens enough to leave a trail when you scrape the bottom of the pan. Stir in 1/2 cup (8 g) of cilantro and the kosher salt, then remove from heat.
Spread a thin layer of the poblano sauce across the bottom of the prepared baking dish. Place a portion of shredded chicken into each warmed tortilla, roll tightly, and place seam-side down in the dish.
Pour the remaining sauce evenly over the rolled tortillas. Scatter the shredded pepper jack cheese across the top.
Bake uncovered for 15 to 20 minutes. The cheese should be melted and the enchiladas must reach a safe internal temperature of 165°F (74°C).
Squeeze the lime juice over the warm enchiladas and sprinkle with the remaining 1 tablespoon of cilantro before serving.