Poulet au Vin Jaune du Jura (Creamy Braised Chicken with Jura Wine and Morels)

Ingredients

• 1 oz. dried morels (about 1½ cups)
• ¹⁄₄ cup plus 1 Tbsp. all-purpose flour, divided
• One 3- to 4-lb. free-range chicken
• 4 Tbsp. unsalted butter, divided
• 1 large shallot, finely chopped (¾ cup)
• 1 medium garlic clove, finely chopped (1 tsp.)
• 2 cups vin jaune de Jura, divided
• 5 sprigs of fresh thyme
• 1 bay leaf
• 8 oz. (1 cup) crème fraîche
• Kosher salt and freshly ground black pepper, to taste
• Crusty bread, noodles, or rice, to serve