In a large saucepan with a tight-fitting lid, brown ground beef, onion, mushrooms and garlic over medium-high heat; drain off any excess fat.
Add broths, water and pumpkin; stir until well blended, thinning with additional water if needed; season with salt and pepper.
Bring mixture to a boil then reduce heat, cover and simmer for 20 to 30 minutes.
Add milk and simmer for 10 to 15 minutes longer.