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Melt butter over medium-low heat in a large saucepan or frying pan until foaming. Sprinkle in flour while constantly whisking, and cook until raw flavor is gone, about 1 to 2 minutes. Slowly add broth, whisking constantly until mixture is smooth. Add milk and again whisk until smooth. Bring to a simmer over medium heat and cook until slightly thickened, about 10 minutes. Remove from heat, add chicken and vinegar, and stir to coat. Season well with salt and freshly ground black pepper.