Preheat the oven to 350°F (175°C).

In a skillet over medium heat, brown the ground beef and chopped onion until the beef is cooked through and the onion is translucent. Break the beef into small pieces as it cooks.

Drain excess fat and return the meat to the skillet.

Add the sliced carrots, sliced celery, and frozen peas to the skillet and cook for an additional 5 minutes, stirring occasionally until the vegetables are slightly tender.

In a large bowl, combine the cooked beef and vegetable mixture with the cooked egg noodles, cream of mushroom soup, and salt and pepper. Mix gently.

Pour the mixture into a greased casserole dish and sprinkle the shredded cheddar cheese evenly over the top.

Bake for 25 to 30 minutes, or until the cheese is bubbly and golden.

Let the casserole rest for 5 minutes before serving.