Preheat oven to 450°F.
In a Dutch oven, cook beef, mushrooms, onion, garlic, thyme, salt, and pepper over medium-high heat for 25-28 mins, stirring and breaking up meat until liquid evaporates.
Stir in flour, cook 1 min. Add milk, broth, simmer and scrape browned bits until thick, about 3 mins. Remove from heat, stir in Parmesan. Transfer to a 13x9-inch baking dish.
Top with peas and corn. Arrange tater tots on top in a single layer.
Bake 35-38 mins until tots are golden and filling bubbles, rotating halfway.
Cool for 15 mins before serving.
To make ahead, cool beef mixture, cover, refrigerate up to 24 hrs. Reheat covered, 15-20 mins. Add peas, corn, tots; bake as directed.