Preheat the oven to 350F. Grease a 2-quart casserole dish; set aside.

In a large skillet, melt 1 tablespoon of the butter. Add in the onion, carrot and celery and cook until the vegetables are starting to soften. Remove from skillet to a bowl.

In the same skillet, melt the remaining 2 tablespoons of butter. Whisk in the flour and cook for a minute or two. Slowly whisk in the chicken broth and the milk. Season to taste with salt and pepper. Cook until thickened.

Remove from heat and stir in the sour cream, pimientos, chicken, and cooked vegetables. Pour into the prepared baking dish. Sprinkle with 3/4 cup of the shredded cheddar cheese. Place in the oven and bake for 15 minutes.

Meanwhile, in a large bowl, combine the flour, baking powder, sugar, salt and baking soda. Cut in the butter with a pastry cutter. Stir in the buttermilk. The mixture will be very sticky. Using a spoon or an ice cream scoop, make 9 balls out of the mixture, dropping each into the flour and coating the outside.

When the chicken gravy is done cooking, remove it from the oven and increase the oven temperature to 450F. Drop the biscuits on top. Brush the tops of the biscuits with the melted butter, then sprinkle the remaining 1/4 cup of shredded cheddar cheese over the top.

Return to the oven and cook until the biscuits are cooked through, 15-20 minutes.