Sprinkle chicken pieces with salt and pepper. Preheat a Dutch oven or big pot over medium heat. Melt ⅔ of the butter, olive oil, and then the chicken skin side down. Cook 3 minutes on each side, being careful not to burn the skin. Remove to a plate. Do in 2 batches if overcrowded.
Add chopped onion, celery, and mushrooms and cook for 3 minutes then add garlic and stir. Scatter flour into pot and stir until absorbed by vegetables, about 1 minute.
Deglaze pan with vermouth for about 45 seconds, then add broth, thyme and bay leaf followed by accumulated chicken juices then lay chicken pieces over vegetables, skin-side up. Cover and cook over medium-low heat for 30 minutes.
While the chicken's cooking, microwave carrot slivers in a partially sealed plastic bag for 1 minute, then drain. Microwave leeks the same way for 3 minutes. Drain and combine with carrots.
Stir carrots, leeks, lemon juice and cream into the pot with the chicken. Bring to a simmer and pour into a large bowl to serve.