In a 3- to 4-quart pan, mix broth, milk, and polenta. Stir over high heat until boiling; reduce heat to low and stir often until polenta is smooth to taste, 12 to 15 minutes. If polenta is thicker than desired, stir in 2 to 4 tablespoons water.
Meanwhile, pour oil into a 5- to 6-quart nonstick pan over high heat, tilting to coat bottom. When oil is hot, add sausages and break apart with a spoon. Add garlic and chili flakes; stir often until meat is browned and crumbled, about 4 minutes. Add mustard greens and stir often until wilted, about 2 minutes.
Spoon polenta equally into 4 bowls. Top equally with sausage-mustard green mixture. Sprinkle with cheese and add salt and pepper to taste.