Prepare the Pan: Grease a 13 x 9-inch pan generously with butter, then set it aside.
Mix the Egg Mixture: Whisk the egg and milk together in a small bowl until well combined, then set aside.
Boil the Water: Pour the water into a large pot and bring it to a boil.
Cook the Grits: Add the sea salt to the boiling water, then gradually pour in the grits while whisking continuously to prevent lumps.
Thicken the Grits: Cook the grits for 5 minutes, stirring from time to time, until they become very thick.
Temper the Egg Mixture: Whisk about 1 tablespoon of the hot grits into the reserved egg-and-milk mixture until thoroughly combined.
Combine the Mixtures: Slowly whisk the tempered egg mixture back into the pot of grits until everything is well blended.
Add the Cheese: Stir the shredded cheddar cheese into the hot grits until evenly incorporated.
Chill the Grits: Transfer the cheesy grits to the buttered pan and use a spatula to spread them into an even layer. Allow them to cool to room temperature, then refrigerate until completely cool and firm, at least 2 hours or preferably overnight. Cut the Grit Cakes: Bring the chilled pan back to room temperature, then cut the firm grits into 8 squares.
Heat the Butter: Add 2 tablespoons of butter to a large nonstick skillet and heat over medium-high heat until the butter has melted.
Cook the Grit Cakes: Place the grit cakes in the skillet and cook until golden on both sides, about 4 minutes per side.
Serve Warm: Transfer the golden grit cakes to serving plates and serve warm with additional butter on top.
