Preheat the oven to 425°. In a large Dutch oven or enameled cast-iron casserole, melt 2 tablespoons of the butter. Add the shallots, mushrooms and carrot chunks and cook over moderate heat, stirring, until the shallots and mushrooms are softened, about 8 minutes. Add the wine and cook until completely evaporated, about 1 minute. Stir in the 1 tablespoon of flour and the broth and bring to a boil. Season with salt and pepper. Simmer until thick, about 3 minutes. Stir in the chicken and peas.
In a food processor, combine the remaining 1 1/4 cups of flour with the remaining 4 tablespoons of butter and the chopped sage and thyme; pulse 5 times. Add the milk; pulse just until a soft dough forms.
Using a small ice cream scoop or a tablespoon, scoop 20 balls of dough over the chicken stew. Bake in the center of the oven for 25 minutes. Turn on the broiler and broil for 1 to 2 minutes, until the biscuits are golden. Let rest for 5 minutes before serving.