Step 1

Place a rack in middle of oven; preheat to 400°. Melt 1 Tbsp. unsalted butter in a large skillet or medium Dutch oven over medium-high heat. Add 1 lb. ground beef (preferably 20% fat), 1½ tsp. Diamond Crystal or 1 tsp. Morton kosher salt, and ½ tsp. freshly ground pepper and cook, stirring and breaking up meat into smaller pieces with a wooden spoon or heatproof rubber spatula, until browned and cooked through, about 5 minutes; remove pan from heat. Using a slotted spoon, transfer meat to a 13×9″ baking dish. Pour off all but 1 Tbsp. fat in pan; discard. (Do not wipe out any of the browned bits.)

Step 2

Melt 1 Tbsp. unsalted butter in same pan over medium-high heat. Add 1 lb. zucchini, summer squash, or white button mushrooms (depending on the season), coarsely chopped, ½ large onion, coarsely chopped, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and ½ tsp. freshly ground pepper and mix to combine, then cook, mostly undisturbed, until vegetables starts to brown, about 5 minutes. Continue to cook, stirring often, until tender, 8–10 minutes more. Transfer to baking dish and stir into meat. Spread into an even layer.

Step 3

Melt remaining 4 Tbsp. unsalted butter in same pan over medium heat. Add 1 large carrot, peeled, coarsely chopped, 1 large celery stalk, coarsely chopped, 3 garlic cloves, finely chopped, ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt, ½ tsp. freshly ground pepper, and remaining ½ large onion, coarsely chopped. Cook, stirring occasionally, until vegetables are tender and browned slightly, 8–10 minutes. Sprinkle ⅓ cup (42 g) all-purpose flour evenly over and cook, stirring constantly, until no pockets of dry flour remain, about 3 minutes.

Step 4

Stirring constantly, gradually add 2½ cups whole milk, then 1½ cups low-sodium vegetable broth and 1 Tbsp. soy sauce. Cook, stirring often, until sauce thickens, about 3 minutes. Remove from heat and stir in ¼ cup finely chopped parsley. Taste sauce and season with more salt and pepper if needed.

Step 5

Top meat mixture with 2 cups frozen corn, green beans, and/or peas, then pour sauce over. Scatter 4 oz. cheddar, coarsely grated (about 1 cup), evenly over and arrange one 32-oz. bag frozen tater tots in a single layer on top, pressing them in slightly to adhere. (There may be extra depending on how you arrange them.)

Step 6

Bake hotdish until bubbling and tater tots are browned on top, 45–50 minutes. Season with salt and pepper and let cool 10 minutes. Top with more finely chopped parsley just before serving.