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Bring the stock to a light boil and set aside.
Roughly chop the chicken carcass and/or wings.
In a stockpot or large heavy-bottomed saucepan, heat the oil over medium-high heat and sear the chicken pieces for 10 to 12 minutes, or until the meat is a deep brown but not burnt. Scoop out the meat pieces and set aside. Remove any excess fat from the pot.
Lower the heat to medium, and stir in the carrot, onion, celery, garlic, and mushrooms, and cook for 2 minutes. Add the tomato paste, thyme, and bay leaf and stir well. Pour in the wine and deglaze by scraping the bottom of the pot with a wooden spoon to detach and blend the caramelized juices into the liquid. Reduce until roughly 2 tablespoons (30 ml) of liquid remain.
Return the chicken pieces to the pot and stir in the toasted flour before pouring in the stock and adding the salt and peppercorns.
Wait for the liquid to come to a boil, then simmer for 30 minutes to reduce the liquid by roughly half, occasionally removing any foam that forms on the surface.
Remove the pot from the heat and let the demi-glace cool for 15 minutes before straining it through a fine-mesh sieve into a bowl.
The demi-glace can be used immediately, kept in the fridge for 3 days, or frozen for several months.