Pat the chicken thighs dry. Season with salt and pepper.
In a Dutch oven or high-sided skillet, heat oil over medium-high heat. Cook chicken until golden brown, 3-4 minutes per side. Remove from pan; keep warm.
In same pan, sauté the mushrooms, carrots, tomato paste and remaining ½ tsp. salt for 2 minutes. Add the broth cognac and wine; bring to a boil. Return chicken to the pan; reduce heat. Cook until chicken reaches 170° and carrots are just tender (about 40 minutes), add in peas and pearl onion mixture (Oignons Glacés a Brun recipe below) at about 30 minutes and finish 8-10 minutes. If desired, top with chopped fresh thyme.
Serve over mashed potatoes.
When the butter and oil are bubbling the skillet, add the onions and sauté over moderate heat for about 10 minutes, rolling the onions about so they will brown as evenly as possible. Be careful not to break their skins. You cannot expect to brown them uniformly.
Pour in the liquid, season to taste, and add the herb bouquet. Cover and simmer slowly for 40 to 50 minutes until the onions are perfectly tender but retain their shape, and the liquid has evaporated. Remove the herb bouquet. Serve them as they are.