Preheat the oven to 400 degrees. Melt 3 tablespoons of butter in a large skillet over medium heat. Add the onion, carrots, and celery. Cook until the vegetables soften, about 5 minutes.

Sprinkle 3 tablespoons of flour over the vegetables and stir for 1 minute. Slowly pour in the chicken broth while stirring to avoid lumps. Add the milk and bring the mixture to a simmer until it thickens.

Stir in the shredded chicken and frozen peas. Remove the skillet from the heat and pour the filling into a 9-inch baking dish.

In a mixing bowl, combine 1 1/2 cups flour, baking powder, and salt. Cut in the cold butter cubes until the mixture looks like coarse crumbs. Pour in the buttermilk and stir just until a dough forms.

Drop spoonfuls of the dough over the filling. Bake for 25 to 30 minutes until the crust turns golden and the filling bubbles at the edges.