This dish goes back to my very first day of cooking school years ago. I have been in love with it ever since. It's all about crisping the chicken skin, and then braising it in an aromatic liquid, combining the best of roasting and slow cooking.
Preheat oven to 350 degrees F. Put the olive oil in a large, heavy-bottomed braising pan over medium-high heat. Generously season both sides of chicken legs with salt and pepper. When the oil is slightly smoking, add chicken legs and brown, about 5 minutes per side.
While chicken is browning, peel and chop the carrots; dice the celery and onion. (All the pieces should be about the same size.) Make sure the chicken is browning evenly on both sides.When the chicken is nicely browned, remove from the pan and set aside. Add vegetables to the pan, stir to combine, and reduce heat to medium. Next, peel and lightly smash the garlic; add to pan. Sweat the vegetables for 5 minutes, being careful not to brown. Season lightly with salt and pepper.
Add tomato paste, raise the temperature to medium-high, and stir to incorporate; the tomato paste will begin to stick to the bottom of the pan as it browns. Deglaze pan with cognac in two additions (being careful for flare-ups!). Let the alcohol cook off until the pan is almost dry, stirring the vegetables and scraping down the sides of the pan to prevent burning. Add white wine and allow it to reduce. Add parsley, bay leaves, thyme, and rosemary to the pan; then spread the chicken pieces on top in an even layer. Pour in just enough chicken broth to cover the meat halfway without submerging. Bring to a simmer, then cover the pan and place in the preheated oven for 35–40 minutes.
Remove from oven and place chicken legs on a rimmed serving platter. Strain cooking liquid into a small saucepan, reserving vegetables. Over medium to high heat, reduce liquid at a simmer until it is thick enough to thinly coat a spoon. Taste, and adjust seasoning as needed. To plate: Place chicken on a plate and add some reserved vegetables. Pour reduced sauce generously over each serving. Garnish with torn parsley leaves and serve.