Sear the 1/2 cup lardons or thickly sliced bacon in the 2 tablespoons olive oil in a heavy-bottomed pot over medium-high heat until lightly browned, about 2 minutes. Transfer to a plate, leaving the drippings in the pot.
Add the 3 1/2 to 4 1/2 pounds chicken, being careful not to crowd the pieces. You may need to work in batches. Sear the chicken, turning occasionally, until nicely browned on all sides. (If working in batches, return all the chicken to the pot.)
Carefully pour the 1/4 cup Cognac into the pot and wait until it becomes bubbling hot. If desired—and if you're brave—ignite the sauce with a match. Let it flame for a minute, gently tilting the pot by its handle and swirling the sauce to burn off the alcohol. To extinguish the flames, simply cover the pan with its lid.
Add the 1 bay leaf and 1/4 teaspoon dried thyme to the pan, nestle the 20 pearl onions around the chicken, and season with salt and freshly ground black pepper. Cover the pot and simmer gently, turning the pieces once for about 10 minutes.
Uncover the pot, sprinkle the 3 tablespoons all-purpose flour over everything, and turn the chicken and onions so the flour is absorbed by the sauce. Cover and cook, turning once or twice, for 3 to 4 minutes more.
Remove the pot from the heat and gradually stir in the 2 cups red wine and enough of the homemade chicken stock to almost cover the chicken. Add the lardons, 2 garlic cloves, and 1 tablespoon tomato paste, cover, and simmer gently for 25 to 30 minutes. Test the chicken for doneness. (There should be no trace of pink, and the juices should run clear when the meat is pierced with a knife.) Transfer the pieces to the plate when they're done.
If the onions aren't quite tender, continue cooking them in the sauce, then return the chicken to the pot, add the 3/4 pound fresh mushrooms, and simmer for 4 to 5 minutes.
The sauce should be just thick enough to coat the chicken and vegetables lightly. (If the sauce is too thin, bring it to a boil and reduce it to the desired consistency. If the sauce is too thick, thin it with spoonfuls of stock.) Taste and correct the seasoning accordingly.
Serve immediately, garnished with fresh thyme sprigs.