Preheat oven to 350 degrees.
Whisk together flour and water until smooth and lump-free. You'll have something that is about the consistency of pancake batter.
Dice the 2 pounds peeled potatoes into 1/4 inch cubes, and whisk the 6 eggs along with salt and pepper to taste.
In a heavy skillet on medium high heat, add ground sausage and cook until completely browned and very crispy. It should look very dark in areas, and have a lot of crisp texture to it. The crispier the better, (without burning). Remove from heat, drain excess grease.
Add the 4 cups milk, flour slurry (1/2 cup flour+ 3/4 cup water), and salt and pepper. Return to heat and simmer on medium until the gravy bubbles and thickens. Taste that salt and pepper level is to your liking. Turn off heat.
Grease a 13x9 inch, or equivalently large casserole dish with either a small amount of butter, or a bit of the gravy. Layer the diced potatoes first, in an even layer, press down into the dish.
Pour beaten eggs evenly over potatoes.
Sprinkle with half of the shredded cheese.
Spoon on the sausage gravy over the cheese, in an even layer.
Top with remaining shredded cheese. If it looks pretty full, put it on a baking dish, to catch any overflow or drips.
Bake in a 350 degree oven for 45 minutes to 1 hour, until potatoes are completely tender. Test by piercing them with a fork. If you notice that the cheese is getting too dark, cover with foil or a lid, and continue to bake until done.
If you are making this ahead, complete the entire assembly instructions; cover and freeze. Frozen casserole will need to add 20-30 minutes cooking time.