Preheat to 450°F; cook beef, mushrooms, onion, garlic, thyme, salt, and pepper until liquid evaporates, 25–28 min.
Stir in flour; cook 1 min.
Add milk and broth; simmer 3 min; stir in Parmesan; transfer to baking dish.
Top with peas, corn, and tater tots; bake 35–38 min until golden and bubbly; cool 15 min.
For make‑ahead, cook filling, cool, refrigerate up to 24 hr; then bake with toppings as directed.