Preheat the oven to 375 degrees.
In a pan over medium heat, add the olive oil and heat. Then add in the onion, garlic and poblano peppers. Cook for 5 minutes until they begin to soften.
Next add the butter and cook for another 5 minutes, stirring on a regular basis.
Then stir in the flour and make sure the pepper mix is coated so the flour begins to create a thick base.
Add in the chicken broth and mix and/or whisk to ensure there are no lumps. Let simmer 5 minutes.
Add in the milk and bring the mixture to a simmer again. Don't let it boil and continuously stir.
Turn off the heat and use the immersion blender to create a creamier sauce. You don't have to blend all the vegetables, but you do want to blend them about halfway down.
Turn the heat back on low and add the lime juice. Bring to a simmer again and add in the salt. Additional salt and pepper as desired. The sauce should nicely coat the back of a spoon now. When it does, turn off the heat again and add in the cilantro.
Place enchilada shells in a baking pan and add a few spoonfuls of chicken into each. Sprinkle half the cheese into the shells.
Once rolled, coat the enchiladas with the creamy poblano sauce. Add the rest of the cheese over the top and bake for 15-20 minutes.
Serve immediately with additional cilantro.