Preheat the oven to 400 degrees.
To make the creamed soup: In a large pot, melt the butter over medium-high heat. Add the carrots, celery, onion and a pinch of salt and cook, stirring, until soft, about 12 to 15 minutes.
Add the mushrooms, if using, and stir 2 to 3 minutes more. Stir in the flour and cook for another minute. Stir in 1 1/2 cups milk and cook, stirring, until thickened. Stir in the remaining 1 1/2 cups milk and cook, stirring, until very thick. Add the bouillon and combine, then season with salt and pepper to taste. Pour into a bowl and set aside.
To make the hot dishes: Wipe out your pot and set it back over medium high heat. Add the ground beef and cook, breaking up with a wooden spoon, until browned. Season with salt and pepper. Stir in the peas, corn or green beans. Using a slotted spoon to drain the juices, transfer half of the beef mixture to 3-inch ramekin dishes and spread it out evenly. Layer on half the soup mixture, then the other half of the beef, then the other half of the soup. Top with a thin layer of the shredded cheese. Cover with a layer of Tater Tots, and bake until the tots are golden brown and crispy on top, about 45 minutes. Let cool slightly, top with parsley if desired, and serve with ketchup.