Preheat oven to 375 degrees Fahrenheit
In a large dutch oven or heavy bottom skillet heat to medium heat. Once butter has melted, add in diced onion and cook until translucent (3-5 mins).
Add in salt, pepper, garlic, parsley and flour and stir until pasty. Whisk in chicken broth and beer to onion mixture. Bring mixture to a simmer (5-10) minutes. Sauce will have thickened. Add in half and half and stir. Mix sauce with potatoes, canned green chilis, carrots and chicken. Transfer to a casserole dish and bake for 30-40 minutes or until potatoes are tender.
While casserole mixture is baking, prepare dumpling batter. In a large bowl whisk together all purpose flour, baking powder, sugar and salt. With a pastry blender or your hands blend together the flour and butter until mixture is crumbly. Slowly stir in the milk while gently folding the batter until it comes together.
Drop 8-10 spoonfuls on top of casserole and bake for an additional 15-20 minutes or until the dumplings are golden and a tooth pick removes clean.