Prepare the broccoli-cheddar tots according to directions. Bake them for 10 minutes instead of 20 minutes as you will finish them in the oven with the rest of the dish.
Heat a large oven safe skillet over medium heat and melt the butter. Add chicken, onion, celery, and carrots. Sauté until the veggies are starting to soften, about 5 minutes. Add minced garlic, cooking for an additional minute. Sprinkle the flour over the vegetables and stir. Pour in the chicken stock and half & half.
Add the broccoli, salt, pepper, thyme, and parmesan cheese. Cook over medium heat until the broccoli is tender-crisp, 5-7 minutes. Top with broccoli-cheddar tots and place in the preheated oven for 10-15 minutes until the tots are crisp and slightly browned.