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Preheat the oven to 425°. In a large ovenproof nonstick skillet, melt 2 tablespoons of the butter. Add the onion, mushrooms and carrots and season lightly with salt and pepper. Cover and cook over high heat, stirring once, until the vegetables are just softened, about 1 minute. Uncover and cook, stirring frequently, until lightly browned, about 5 minutes. Stir in the flour and paprika and cook, stirring, for 1 minute. Add the stock and Madeira and cook, stirring, until blended. Add the milk and bring to a gentle boil. Stir in the chicken and peas and season with salt and pepper. Remove from the heat.
Arrange the bread over the chicken mixture, trimming it to fit snugly in a single layer. Brush the bread with the remaining 2 tablespoons of butter. Bake for about 20 minutes, until the filling is bubbling and the bread is golden. Serve right away.
Wine Pairing: Beaujolais, made with the Gamay grape, is light-to-medium-bodied and low in tannins; it's flexible enough to pair with all kinds of dishes, including this earthy potpie. Pour the 2008 Château Thivin Côte de Brouilly or the 2008 Georges Duboeuf Morgon.