In a large soup pot over medium-high heat, melt the butter. Add onion and cook, stirring frequently for 3 minutes, until softened. Add the garlic and continue cooking for an additional 30 seconds.

Whisk in the flour, Italian seasoning, and black pepper; the mixture will be thick. Gradually add the chicken-style broth, whisking continuously to remove any clumps.

Stir in the hash browns and broccoli. Bring to a boil then reduce the heat to low. Simmer for 10 minutes, until heated through.

Add the milk and cheese. Keeping the heat on a low simmer, cook for about 5 minutes, stirring occasionally until the cheese is melted. Adjust seasoning to taste and serve hot.