Lightly spray a 4 to 6 quart slow cooker and add pork shoulder.
Whisk Coca-Cola, brown sugar, apple cider vinegar, liquid smoke, garlic powder, minced onion, salt, pepper, paprika, and ginger together. Pour over pork.
Cover and cook on Low for 10 hours, until pork shreds easily.
Remove pork to a cutting board and shred with two forks. Reserve about 2 cups plain shredded pork for enchiladas.
Melt butter in a saucepan over medium-low heat. Whisk in flour and cook 1 to 2 minutes.
Slowly whisk in chicken broth and cook 3 to 5 minutes, until thickened and smooth.
Remove sauce from heat and cool 5 minutes, then stir in sour cream and diced poblano until smooth.
Preheat oven to 350°F and spray a 9x13 baking dish.
Fill tortillas with reserved pork, roll tightly, and place seam-side-down in the baking dish.
Pour poblano cream sauce over enchiladas and sprinkle with cheese.
Bake uncovered at 350°F for 20 to 25 minutes, until cheese is melted and sauce bubbles at the edges.
Optional: broil 2 to 3 minutes for a lightly browned top, watching closely. Rest 5 minutes and garnish with green onions.
Notes
Let the cream sauce cool for 5 minutes before stirring in sour cream so it stays smooth and does not break. Reserve plain shredded pork for the enchiladas before adding any barbecue sauce or extra cooking liquid. Pork shoulder or butt works best because lean pork can dry out over 10 hours. Assemble up to 24 hours ahead and add 10 to 15 minutes to the bake time if baking cold.