Season chicken liberally with salt and pepper, then with dried herbes de Provence. Heat oil in the Le Creuset braiser over medium-high heat, then brown chicken on both sides to a deep golden brown. Remove the braiser from the heat, remove chicken to a plate, and spoon off any excess fat from pan.
Add remaining ingredients to the pan, scraping any brown bits from the bottom. Return the chicken to the pan and bring to a simmer. Cover with the lid and cook in the oven for 1.5 hours, basting every 30 minutes.
Remove from the oven and transfer the chicken, shallots, and garlic to a warm platter. Discard thyme sprigs from the pan and reduce sauce if necessary. Small curds will form in the sauce; don't worry, they're delicious!
Serve the chicken with sauce and a few pieces of shallots and garlic.
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