In a Dutch oven or large saucepan, melt the butter over medium heat.

Add the onion, celery, carrot, and garlic and sauté until everything is starting to soften and the onions are becoming translucent, about 5 minutes.

Add the ground beef and break it up as it cooks. Cook until it's no longer pink, about 8 minutes.

Add in the broth and potatoes and mix thoroughly. Turn up the heat to medium high, then bring to a boil. Once it boils, turn the heat back down to medium, letting it boil and simmer for about 6-8 minutes, or until the potatoes are softened.

Note that if your pan doesn't hold heat well, only turn it down to the notch above medium. You just want to make sure it stays boiling at a nice simmer.

Once the potatoes are done, turn the heat to low and add the milk in and mix, then add the cheese slowly, stirring to melt each addition. Finally, add the dill and mix and serve!