Place large skillet on stovetop over medium heat. When skillet is warm, add 4 tablespoons unsalted butter and let butter melt completely, swirling and tilting skillet as needed to distribute oil across entire pan.
When butter has melted, add ½ cup diced celery, ½ cup finely chopped carrots, ½ cup diced yellow onion, ½ cup frozen peas, and 2 small red potatoes to skillet. Stir to incorporate.
Sauté vegetables until just tender but still crisp, approximately 5 minutes. Stir often.
When vegetables are tender, sprinkle 4 tablespoons all-purpose flour over vegetables, then add 1 teaspoon garlic powder, 1 tablespoon dried thyme, 1 teaspoon dried whole sage leaves, 1 pinch nutmeg, salt, and freshly cracked black pepper. Stir to coat vegetables completely in spices and flour.
Cook vegetables 1 minute, stirring constantly. After 1 minute, remove skillet from heat and let cool slightly.
When pan has cooled, slowly add 1 cup whole milk (at room temperature) and 2 cups low-sodium chicken broth to skillet in small increments, stirring well after each addition. Once all milk and chicken broth has been added, return skillet to heat.
Add 2 cups chopped, roasted chicken and stir to incorporate. Taste mixture and add salt and/or pepper if desired. When satisfied with flavor, transfer mixture to prepared baking dish and set aside.