Melt the butter in a large pot or Dutch oven over medium heat. Add the chopped onion, diced carrots, and celery. Cook until the vegetables soften, about 5 to 7 minutes.
Sprinkle the flour over the cooked vegetables. Stir constantly for 1 minute to cook out the raw flour taste, creating a roux.
Gradually whisk in the chicken broth, ensuring no lumps form. Bring the mixture to a simmer.
Stir in the dried thyme and rosemary. Let the soup simmer gently for 10 minutes, allowing it to thicken slightly.
Add the shredded chicken and the frozen peas and carrots mix. Cook until the vegetables are tender, about 5 more minutes.
Reduce the heat to low. Stir in the heavy cream and milk. Heat through, but do not boil after adding the cream. Season with salt and pepper.
While the soup simmers, prepare the biscuit toppers according to package directions. Bake until golden brown.
Ladle the soup into bowls and top each serving with one or two warm biscuit toppers.
Notes
For a gluten-free option, substitute the flour with 3 tablespoons of cornstarch mixed with 1/4 cup cold water (a slurry) added after the broth, just before simmering.
You can use rotisserie chicken for quicker preparation.
If you prefer a richer flavor, substitute half of the chicken broth with half-and-half.