In a 12- skillet or braiser, brown the ground beef and shallot. Break up the ground beef as it cooks.
1 pound ground beef, 1 medium shallot
Meanwhile, cook the gnocchi and broccoli florets according to the instructions on the gnocchi - usually 3 to 4 minutes. Don't overcook the gnocchi because it'll keep cooking once you add it to the beef mixture. When cooked, drain and reserve 1/2 to 3/4 cup of the water.
1 head broccoli florets, 16 ounces gnocchi
When the ground beef is cooked through, remove it from the pan with a slotted spoon. Add the butter to the grease in the pan and sprinkle the flour over it. Cook for 1 to 2 minutes, but don't toast the flour (don't let it get dark). Stir in salt and pepper.
Slowly add the milk to the pan, whisking as you add it. Keep stirring until it is smooth. Bring the sauce to a low boil, stirring to make sure it doesn't stick to the bottom. Boil for 2 minutes.
2 and ½ cups whole milk, ½ teaspoon salt, 1 teaspoon Ground black pepper
Turn the heat to simmer. Add the ground beef, gnocchi, and broccoli to the pan and stir to coat them with the sauce. Cover and simmer for 5-10 minutes. (You can cover it with 1 to 2 cups of mozzarella cheese and cook it until the cheese melts.)
Sprinkle freshly grated Parmesan cheese over the top and serve.