Preheat the oven to 425 degrees.
In a large pot set over medium heat, melt the butter. Add the carrots and celery, sauté for 5 minutes. Add the onion and garlic, sauté until tender and transparent, about 3 minutes.
Sprinkle flour over top and whisk to combine. Cook for 1 minute. Slowly pour in the chicken broth, whisking vigorously to help thicken without clumps forming. Add half and half along with the seasonings, whisk to combine. Bring mixture to a simmer, continue cooking for 15 minutes or until the filling is nice and thick (remember it has to be thick enough to hold up the weight of the biscuits).
Meanwhile, to make the biscuits, in a large mixing bowl, combine flour, baking powder, cornstarch, sugar, and salt. Whisk to combine. Add the cold cubed butter. With a pastry blender, cut the butter until pea-size clumps for. Pour in 1 1/2 cups of buttermilk and mix until the dough becomes moist and no loose pockets of flour remain. You can either roll out and cut 12 biscuits or divide the dough into 12 pieces and form 12 "rustic" biscuits by hand. Set aside.
Add the chicken, peas, and corn. Transfer filling to a 9x13-inch baking dish and top with biscuits. Brush the tops of the biscuits with the reserved buttermilk.
Place in oven to bake for 20-25 minutes or until the tops of the biscuits are golden brown. Remove from oven and let stand for 10 minutes before serving.