Granny's Chicken & Dumplings

Ingredients

• 5 tablespoons cold unsalted butter, divided
• 1 small yellow onion, minced
• 1 stalk celery, thinly sliced
• 2 carrots, medium-diced
• 1 1/4 cups all-purpose flour, divided
• 3 cups chicken broth
• 2 cups chicken, cooked and diced
• 1/3 pound green beans, trimmed, cut into 1-inch pieces
• table salt, to taste
• black pepper, to taste
• 1 teaspoon baking powder
• 1 teaspoon coarse salt
• 2 tablespoons fresh parsley, chopped
• 1/2 cup whole milk
• fresh chives, optional, to taste, for garnish