Set an extra-deep 12-inch nonstick skillet that has a lid over medium heat. Add the sausage to the hot skillet and stir frequently to break it into pieces. While the sausage releases its fat, peel and chop the onion, adding it to the skillet as you chop. Seed and chop the bell pepper, and add it to the skillet. Coarsely chop the sliced mushrooms and add them. After all the vegetables are added, cook until they are softened and the sausage is almost completely cooked, 2 minutes more.

If the hash browns are frozen in big clumps, bang the bag on the counter to loosen; or, after measuring out the amount you're using, rinse under running water in a colander and drain. Scatter the hash browns over the sausage and vegetables and add the water. Cover and cook for 4 minutes. Uncover and raise the heat to medium-high. Cook, stirring often, until the potatoes start to brown, 7 to 8 minutes more, reducing the heat if the potatoes begin to overbrown. Season with salt and black pepper.

Sprinkle the cheese evenly over the mixture, cover, and cook until the cheese melts, about 1 minute, then serve.