Grease a 9x12-inch baking pan with baking spray. Set aside.
Add water, milk, bay leaf, and salt to a large saucepan. Bring to a boil.
Gradually whisk in the grits until you have added them all to the pot. Keep stirring with a whisk to prevent lumps. You may need to remove the saucepan from heat while getting rid of lumps.
Reduce heat to low and cook grits at a bare simmer, covered, frequently stirring, until water is fully absorbed, and grits thickened. This will take you about 15 minutes.
Then remove grits from heat; add butter and cheese. Stir with a whisk until cheese completely melts.
Spread out the grits onto the prepared baking sheet and, using an offset rubber spatula, spread evenly across the bottom of the pan. Let the grits sheet pan sit out until they have cooled to room temperature. Cover with foil and refrigerate until very firm, about 2 hours to overnight.
Heat butter in a large skillet over medium heat when chilled grits are ready to fry. Cut the grits into preferred circular sizes using a cookie cutter. (I used a 2-inch cookie cutter.)
Whisk the egg in a small dish and brush on grit cake tops. Fry each side of grits until golden in color, about 2-3 minutes on each side. Do not forget to brush the other side of the grits cake with the egg wash when you flip it over.
Serve immediately with shrimp or ham.