Cut the poulard into four parts. Peel the 12 small onions and halve them.
Heat 40 g of the butter in a large roasting pan over medium-high heat. Add the poulard pieces and brown them vigorously on all sides for 3-4 minutes per side until golden. Add the halved onions and brown for 2-3 minutes. Peel the garlic clove, chop it, and add to the pan. Pour in the 8 tablespoons of brandy, then add the 500 ml of red wine. Season with salt and pepper to taste. Cover the pan and braise for about 40 minutes until the poulard reaches 74°C / 165°F at the thickest part of the meat.
While the poulard braises, rinse, clean and halve the 250 g of mushrooms. Heat the remaining 20 g of butter in a separate pan over medium heat. Add the mushrooms and steam for about 3 minutes until tender. Season with the 1 teaspoon of rubbed thyme and transfer to the braised poulard.
Stir the 2 tablespoons of sauce thickener directly into the braising liquid to thicken the sauce. Season to taste with salt and pepper. Scatter the fresh thyme leaves over the top and serve hot.