Preheat oven to 400°F (200°C).
Poach chicken breasts in 3 cups of chicken broth for 8-10 minutes, or until fully cooked. Remove, cool, and shred.
In a large pot over medium heat, melt 1 tablespoon of butter. Sauté the onions, celery, carrots, and garlic until softened, about 5 minutes.
Stir in onion powder, thyme, rosemary, and sage. Cook for another minute until aromatic.
Sprinkle in flour and stir continuously for 1-2 minutes to form a roux.
Gradually add the remaining 2 tablespoons of poaching broth while stirring. Whisk in the half and half or milk to create a smooth sauce.
Crush the chicken bouillon cube into a fine powder and mix into the sauce, along with the soy sauce. Add the shredded chicken and mix well.
Transfer chicken mixture to an oven-safe 9×13-inch baking dish. Let the mixture cool slightly before topping with biscuit dough.
For biscuit topping: In a bowl, combine flour, sugar, and baking powder. Cut in the cubed butter until the mixture resembles coarse crumbs.
Beat in the eggs and slowly add the milk until a soft dough forms. Knead lightly, then roll out to 1/2-inch thickness.
Cut dough into rounds or rectangles and place over filling. Brush with melted butter.
Bake for 25–30 minutes, or until the top is golden and bubbly. Let cool slightly before serving.