Remove seeds and veins from hatch green chilies and jalapenos (or leave them if you like it spicy), then dice into ¼-1/2 inch pieces. Dice onion into similar sized pieces. Measure 2 tablespoons of unsalted butter into a non-stick pan on the stove top. Allow the butter to melt and then add diced veggies to the pan. Stir to coat with butter and sauté for 15-20 minutes on medium/low heat.
While veggies are sautéing, place a large stock pot on the stove top and turn heat to low. Add remaining butter and allow it to melt (avoid letting it brown). Once it starts to bubble a little bit, add flour and whisk continuously for 2-3 minutes while the mixture thickens. Add chicken broth and whisk until smooth, then increase heat to medium/high. Add pepper, cumin, lime juice, three quarters of your sautéed veggies, a third of your shredded cheddar cheese and chicken pieces. Stir everything together and bring to a boil. Reduce heat back down to medium/low and allow to simmer for 20 minutes.
Preheat oven to 350 F. Chop cilantro and green onion.
Remove chicken pieces from the stock pot and place them on a cutting board. Use two forks to shred and place in a mixing bowl. Add half of your remaining cheddar cheese (reserving the rest to sprinkle on top before baking), remaining sautéed veggies, and half of your chopped cilantro/green onion. Mix everything together.
Carefully transfer verde sauce mixture from the stock pot to a blender. Blend until smooth.
Cover the bottom of a 9×12″ baking dish with a thin layer of sauce. Portion shredded chicken mixture into tortillas, roll them up and line them in your baking dish. Pour sauce on top of the rolled tortillas and sprinkle with remaining cheese, cilantro & green onion. *Tip: avoid drenching your enchiladas too heavily with sauce before baking to avoid soggy tortillas. If you have a lot of extra sauce you can reserve it to use as a topping.
Place dish in the oven and bake for 25-30 minutes (until the sauce is bubbling). Allow enchiladas to cool for 10 minutes before serving. Add desired toppings and enjoy!