Heat the oil in a pan over medium heat. Season both sides of the chicken with salt and pepper. Sear the chicken until golden brown on both sides, then remove from the pan and set aside.
Melt the butter in the same pan over medium heat. Add the mushrooms, shallots, celery, and garlic. Sauté, stirring occasionally, until mushrooms are tender and browned (about 5 minutes). Season with salt and pepper.
Add the wine and simmer until it mostly evaporates. Then stir in the broth, thyme, rosemary, bay leaf, chicken, and potatoes. Cover with a lid and bring to a low simmer until the chicken reaches an internal temperature of 160°F and the potatoes are tender.
Stir in the milk; return to a simmer, and season with salt and pepper, to taste.
Whisk the flour with an equal amount of water until a paste forms. Slowly add the flour mixture into the simmering pan until the liquid becomes thick and creamy (use more or less flour/water mixture to reach a creamy consistency).
Serve with bread. (Very important: Remove the bay leaf, rosemary, and thyme sprigs before serving.)