Bring a large pot of salted water to a boil. Add the diced carrots and cook for 2 minutes. Add the broccoli and cauliflower florets and cook for 2 more minutes. Drain and immediately transfer to an ice-water bath. Drain well and pat dry.

In a large saucepan, melt the butter over medium heat. Add the chopped mushrooms and sauté for 5-7 minutes until deeply browned and fragrant.

Sprinkle the flour over the mushrooms, stirring constantly for 1 minute. Gradually whisk in the whole milk. Simmer for 3-4 minutes until thick, then remove from heat and stir in the grated yellow cheddar cheese, salt, and pepper.

Preheat your oven to 400°F (200°C). Gently fold the blanched vegetables into the cheese sauce until coated. Spread evenly into a 9x13-inch rectangular ceramic baking dish. Arrange the frozen tater tots in a neat, single layer across the top.

Bake uncovered for 35-40 minutes until the tater tots are highly textured, golden-brown, and the sauce is bubbling. Let rest for 10 minutes, then sprinkle with finely chopped fresh parsley before serving.