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In a large pot, sauté lardons over medium heat in 1 tbsp. (15 ml) olive oil. Set lardons aside on a large plate.
Season chicken pieces with salt and pepper and roll in flour.
In the same pot, brown half the chicken pieces over medium heat, 4 to 5 minutes a side. Repeat with the rest of the chicken pieces. Set aside with lardons.
In the same pot, sauté the mushrooms and onions over medium-high heat for about 5 minutes. Set aside in a bowl.
Add garlic and cook for 1 minute.
Deglaze with Cognac. Add red wine and broth. Bring to a boil. Add chicken pieces and lardons, thym, and bay leaf. Season with salt and pepper.
Cover and simmer for 45 minutes over medium heat until chicken meat detaches easily from the bone. Add mushroom garnish 15 minutes before chicken has finished cooking.
In a small bowl, mix butter and flour until mixture is smooth.
Using a slotted spoon, remove chicken pieces, lardons, and garnish from the sauce. Set aside in a bowl. Bring sauce to a boil over medium heat. Add beurre manié with a whisk until sauce thickens. Return chicken and garnish to pot. Let simmer 5 minutes. Adjust seasoning.