Melt the butter in a large pot or Dutch oven over medium heat.
Add the chopped onion, carrots, and celery. Cook until the vegetables soften, about 5 to 7 minutes.
Sprinkle the flour over the vegetables and stir constantly for 1 minute to cook out the raw flour taste. This creates a roux.
Slowly whisk in the chicken broth until the mixture is smooth. Bring the soup to a simmer, stirring occasionally until it thickens slightly.
Stir in the milk or half-and-half, shredded chicken, frozen vegetables, thyme, and rosemary. Heat through, but do not boil after adding the milk.
Season the soup with salt and pepper to your preference.
While the soup simmers, prepare the biscuit toppers according to the package directions. Bake them until golden brown.
Ladle the hot soup into bowls and top each serving with one or two warm biscuits.
Notes
For a richer soup, substitute heavy cream for the milk.
If you prefer a thicker soup, let the roux cook for an extra minute before adding the liquid.
You can use rotisserie chicken for quicker preparation.
For a gluten-free option, substitute the all-purpose flour with a gluten-free flour blend or use 1 tablespoon of cornstarch mixed with 2 tablespoons of cold water (a slurry) added at the end to thicken the broth.