Melt the butter in a large heavy pot over medium-high heat until hot. Add the carrots, celery and onions and season with 1 teaspoon salt and 3/4 teaspoon pepper. Cook, stirring occasionally, until the vegetables are golden, about 12 minutes. Stir in the flour, and then whisk in the broth and milk and bring to a boil, whisking. Boil until slightly thickened, about 3 minutes. Stir in the chicken. Transfer to a 2-quart baking dish.
Pulse the tots in a food processor with the thyme until medium ground. Scatter over the top of the casserole and bake until the filling is bubbling and the topping is golden and crisp, 20 to 25 minutes. Let stand 10 minutes before serving.