Grease a 13 x 9 inch pan with butter and set aside.
Whisk the egg and milk together in a small bowl and set aside.
Pour water into a large pot and bring to a boil. Add salt and then pour in the grits, whisking the whole time. Cook for 5 minutes, stirring from time to time – you want these grits to be THICK.
Whisk about 1 tablespoon of hot grits into reserved egg-milk mixture until well combined. Slowly whisk this mixture back into grits until well blended. Stir in the cheese.
Transfer the grits into the greased pan, smoothing them out with a spatula so they are even. Let them cool off to room temperature, and then pop them in the fridge until they are cool and firm (at least a few hours, but I like to do this overnight).
When you are ready to make the grit cakes, let the pan come back to room temperature, and cut the grits into squares.
Add 2 tablespoons of butter to a large nonstick skillet and heat over medium high heat until melted.
Add grit cakes and cook until golden on both sides, about 4 minutes per side.