Chef John's Chicken and Dumplings

Ingredients (17)

• + 1 (3 to 3 1/2 pound) whole chicken add to pantry
• + 2 ½ quarts cold water add to pantry
• + 1 large carrot, cubed add to pantry
• ✓ 1 stalk celery, chopped
• + 1 onion, chopped add to pantry
• + 3 sprigs fresh thyme add to pantry
• + 1 bay leaf add to pantry
• + 2 tablespoons all-purpose flour, or as needed add to pantry
• + salt and freshly ground black pepper to taste add to pantry
• + salt and freshly ground black pepper to taste add to pantry
• + ½ teaspoon cayenne pepper, or more to taste add to pantry
• + ½ cup creme fraiche add to pantry
• + ½ cup milk add to pantry
• + 2 teaspoons chopped fresh thyme leaves add to pantry
• + 2 eggs add to pantry
• + 2 cups self-rising flour add to pantry
• ✓ 4 sprigs thyme, for garnish