Preheat the oven to 375 degrees Fahrenheit. Lightly grease a 9x13 inch ceramic or glass baking dish with a tablespoon of unsalted butter or nonstick cooking spray. Set it aside while you begin cooking on the stovetop.

Place a large skillet over medium heat. Add the lean ground beef, breaking it apart with a wooden spoon as it cooks. Allow the meat to sizzle until it is beautifully browned and no pink remains, which usually takes about six to eight minutes. Drain off any excess fat from the skillet, leaving just a touch for flavor.

Add the remaining tablespoon of butter, finely diced yellow onion, and minced garlic directly into the skillet with the browned beef. Cook for about three minutes until the onions become soft, translucent, and fragrant.

Stir in the uncooked long grain white rice, cream of mushroom soup, tomato sauce, milk or beef broth, salt, black pepper, garlic powder, and smoked paprika. Mix thoroughly to ensure the rice is evenly distributed and submerged in the flavorful liquid. Gently fold in the frozen mixed vegetables.

Transfer the mixture from the skillet into your prepared baking dish, spreading it out evenly using a spatula. Sprinkle half of the shredded sharp cheddar cheese over the rice mixture, stirring gently to integrate. Cover the top with the remaining cup of cheddar cheese. If desired, sprinkle the buttery breadcrumbs over the cheese for extra texture.

Cover the baking dish tightly with aluminum foil to trap the steam, which allows the rice to cook until tender. Bake in the preheated oven for 35 minutes. Carefully remove the foil and bake uncovered for an additional 10 to 15 minutes, until the cheese is gloriously bubbly and slightly golden around the edges.

Remove the skillet from the oven and let it rest on a heatproof trivet for 5 to 10 minutes before serving. This brief resting period allows the juices to settle and the sauce to thicken slightly, making it much easier to scoop into neat, comforting portions.