Preheat oven to 375°F.
For the gravy, melt butter in a large cast-iron skillet over medium-high heat. Add the onion and carrots, and cook for a few minutes until they begin to soften. Sprinkle the flour over the cooked vegetables and stir a minute or so to cook. Slowly whisk in the milk and stock. Add salt and pepper, and cook over medium heat for 10 minutes until the mixture has thickened. Once thickened, add in the peas, corn, chicken and parsley. Remove from heat.
For the biscuits, whisk together flour, baking powder, baking soda, salt and pepper in a large bowl. Add in the grated butter and cheese, and lightly toss everything with your hands to fully coat the butter with the flour—you don’t want to break the butter pieces up too much. Add in the dill and buttermilk, and mix just until the dough forms to avoid over-handling. If the mixture is too dry, add a little more buttermilk.
Turn the dough out onto a floured surface and press it out until about 1-inch thick with your hands. Using a 3-inch round cutter, cut out the biscuits until you have no dough left. You should have about 10-12 biscuits.
Place the biscuits on top of the chicken gravy, brush each biscuit with a little buttermilk and top with a little extra cheese.
Bake uncovered for 20-25 minutes, until heated through and biscuits are golden brown.