Pat the Whole Chicken dry and cut into serving pieces; season generously with Salt and freshly ground Black Pepper.
Heat a large heavy-bottomed Dutch oven over medium-high heat and add a drizzle of Olive Oil. Cook the Lardons until crisp and browned, then remove with a slotted spoon and set aside, leaving the rendered fat in the pot.
Lightly dredge the chicken pieces in All-Purpose Flour, shaking off excess. Brown the chicken in batches in the hot fat, adding a bit of Unsalted Butter if needed for color; transfer browned pieces to a plate.
Lower the heat, add the Cognac to the pot to deglaze (carefully flambé if you like) and scrape up the browned bits, then pour in the Dry Red Wine and bring to a boil to reduce slightly.
Return the chicken to the pot and add enough Chicken Stock to come about two-thirds up the pieces. Stir in a spoonful of Tomato Paste, a few sprigs of Thyme, the Bay Leaf and a bundle of Parsley Stems tied together, plus the peeled and halved Carrots and smashed Garlic Cloves. Bring to a simmer, cover, and cook gently until the chicken is tender, about 35–45 minutes.
While the chicken braises, melt more Unsalted Butter with a little Olive Oil in a skillet and sauté the Pearl Onions until golden and tender; remove and keep warm.
In the same skillet, sauté the Button Mushrooms in the butter–oil until they release their juices and brown; season with a pinch of Salt and Black Pepper.
When the chicken is cooked, remove the pieces to a platter and skim excess fat from the sauce. If the sauce needs thickening, whisk together a small knob of Unsalted Butter and a spoonful of All-Purpose Flour and stir into the simmering sauce until it coats the back of a spoon.
Return the chicken, cooked Lardons, sautéed Pearl Onions and Button Mushrooms to the pot to warm through for a few minutes. Taste and adjust seasoning with more Salt and Black Pepper as needed.
Serve hot, sprinkled with chopped Flat-Leaf Parsley for brightness. Coq au Vin is lovely over buttered noodles, mashed potatoes or crusty bread to soak up the sauce.
