Preheat the oven to 375°. Spread the thyme, bay leaves and chamomile in the bottom of a large, deep, enameled cast-iron casserole. Season the chicken inside and out with salt and pepper and set it directly on the aromatics. Rub the top of the bird with 3 tablespoons of the butter.

Roll the pizza dough into a rope that is long enough to encircle the casserole. Press the dough onto the rim of the casserole and cover with the lid, pressing it into the dough. Pinch the dough up onto the lid slightly to seal the pot completely. Roast the chicken in the center of the oven for 1 hour and 10 minutes. Remove the lid, cracking the dough, and pierce the thigh to make sure the bird is cooked through. Replace the lid and let the chicken rest for 5 minutes.

Meanwhile, in a large skillet, melt the remaining 1 tablespoon of butter. Add the brussels sprouts, season with salt and pepper and cook over moderate heat until just wilted. Transfer the brussels sprouts to a platter.

Heat the olive oil in the skillet. Add the garlic and cook over moderate heat until golden, about 5 minutes; discard the garlic. Add the mushrooms, season with salt and pepper and cook over moderately low heat until softened and lightly browned, about 10 minutes. Add the demiglace mixture and simmer until slightly reduced, about 5 minutes.

Transfer the chicken to a cutting board and strain the pan juices into a measuring cup. Discard the fat from the juices, then add the juices to the mushrooms. Remove the chicken skin and slice the breast meat. Cut off the legs, cutting them into two pieces each. Arrange the chicken on the brussels sprouts on the platter, spoon the mushrooms and sauce on top and serve with chunks of the baked pizza dough, for dipping.