Wash the potatoes and peel both potatoes and onions. Set the potatoes on the cutting board and dice them and the onions.

Melt the butter in a large sauce pan and add the cut-up onions. Simmer for about 10 minutes and stir frequently so the onions don't stick or burn. You'll want the onions soft but not browned. Many Irish potato soup recipes call for some kind of green vegetables to be added, such as kale, spinach or turnip greens. Some recipes even call for garlic or chives.

Add the water and the milk to the sauce pan and then put in the diced potatoes. You can use vegetable or chicken stock instead of milk and water but you will need to have 4 cups of stock. It will give the potato soup a slightly different taste but it will still be tasty. Adding bits of bacon to your soup can give it a delicious texture and taste.

Bring the soup to a slight boil. Make sure that you stir the contents of the pan frequently to keep the milk from scorching on the bottom and boiling over. Add the cup of cornmeal and stir it in. You can use a whisk to make sure there aren't any lumps. Some people put the soup into a food processor to be blended. This will make cream of potato soup instead of just plain potato soup. If you want a creamier soup without blending, then pour in 1 cup of heavy cream.

Cook until the soup until it is as thick as you want it, adding salt and pepper to taste. Take the pan off the burner and add a cup of cheddar cheese. Stir until the cheese is melted and then serve.