In a medium Dutch oven or heavy-bottomed stockpot, heat the butter or dairy-free alternative over a medium flame. Add the shallot and garlic and sauté until soft and fragrant, about 3 minutes.
Add the potatoes, chicken breast, stock, salt, and herbs. Stir once to combine and bring to a simmer over medium-high heat. Reduce the heat to low and allow to gently simmer uncovered until the potatoes are fork tender and the chicken is cooked through, flipping the breast once halfway through, 15 to 20 minutes depending on the size of your chicken.
Remove the chicken from the broth and transfer to a cutting board to cool. Discard the herb sprigs.
Meanwhile, transfer the contents of the pot to a stand blender (or alternatively, use an immersion blender directly in the pot). Add the milk or dairy-free alternative and puree until thick and smooth. Taste for seasoning and add more salt as needed.
Once the chicken is cool enough to touch, roughly chop or shred the breast into bite-sized pieces. Stir the chicken pieces into the soup base. Transfer the gluten-free cream of chicken soup to bowls and garnish with fresh pepper, or alternative, store in jars for latter or use it in a gluten-free chicken pot pie or casserole recipe.