Prep vegetables: Finely dice onion, celery, and carrots. Mince garlic. Keep pieces small and uniform for even cooking and smooth blending.
Build flavor base: Melt 4 tablespoons butter in large pot over medium heat. Add onion, celery, and carrots. Cook 8-10 minutes, stirring occasionally, until very soft and translucent.
Add aromatics: Add garlic, thyme, tarragon, and bay leaf. Cook 1-2 minutes, stirring constantly, until fragrant.
Make roux: Sprinkle flour over vegetables. Stir constantly for 2 minutes until flour smells slightly nutty. Don't let it brown.
Add broth: Add chicken broth ½ cup at a time, whisking constantly after each addition to prevent lumps. Bring to gentle simmer, whisking until smooth and beginning to thicken.
Simmer: Reduce heat to medium-low. Simmer 15 minutes until vegetables completely tender. Remove and discard bay leaf.
Add chicken: Stir in shredded chicken. Simmer 5 minutes to heat through and absorb flavors.
Blend: Remove from heat. Use immersion blender to blend completely smooth (or blend in batches in regular blender with towel held over lid). Blend 1-2 full minutes.
Strain: Pour blended soup through fine-mesh strainer into clean pot, pressing with ladle to extract all liquid. This creates signature velvet texture.
Make velvet finish: In small saucepan, melt 2 tablespoons butter. Whisk in flour, cook 1 minute. Whisk in warm broth until smooth and thickened. Add to main soup, whisking to incorporate.
Add cream: Return soup to medium-low heat. Slowly stir in heavy cream and milk. Warm 3-4 minutes without boiling. Add pinch of nutmeg.
Season: Taste and adjust with salt, white pepper, and nutmeg. Should be rich, well-seasoned, and delicately flavored.
Serve: Ladle into warm bowls. Garnish with cream drizzle, chives, parsley, and white pepper. Serve with oyster crackers or crusty bread.