In one quart sauce pot bring water to rolling boil. Add butter and salt. Fold in grits and reduce heat to simmer. Cook until grits are tender stirring occasionally. (Can also use quick cooking grits but change water accordingly).
When cooked remove from heat and stir in sausage, cheese, and pepper. Pour hot grits into sheet pan with 1 inch lip. Smooth out to 3/4" thickness. Cool overnight in refrigerator.
The next morning use a 3" round cookie cutter to cut circles (about 8 patties).
Heat oil in large skillet over medium high heat.
Prepare egg wash: Beat egg and milk together.
Dip patties into egg wash then flour then egg wash then bread crumbs.