Preheat the oven to 450° and position a rack in the upper third. Line the bottom of the oven with foil to catch any drips.
In a large pot, melt 4 tablespoons of the butter in the oil. Add the onion, carrots and parsnips; season with salt and pepper. Cover and cook over moderately high heat, stirring, until the vegetables are softened and lightly browned, 8 minutes. Add the beef, thyme and rosemary and cook over moderately high heat, breaking up the meat with a spoon, until no trace of pink remains and any liquid has evaporated, 10 minutes. Stir in 1/4 cup of the flour and cook for 2 minutes. Stir in 3 cups of the milk and the stock and simmer until thickened, 5 minutes. Stir in the peas and season with salt and pepper. Transfer the mixture to two 9-by-13-inch baking dishes.
Put the remaining 4 cups of flour in a large bowl. Cut in the remaining 1 1/2 sticks of butter until the mixture resembles coarse meal. Add the cheddar. Stir in the remaining 2 1/3 cups of milk until a smooth dough forms. Using a 2-tablespoon scoop, arrange 18 mounds of the biscuit dough on the filling in each baking dish.
Bake both potpies on the top rack of the oven for 40 minutes, until the filling is bubbling and the biscuits are cooked through. Turn on the broiler and broil for 1 minute or until the biscuits are golden. Let the potpies rest for 10 minutes before serving.