In a large heavy-bottomed saucepan, bring 4 cups of water to a boil over medium-high heat. Stir in the grits and return to a boil. Season with 1 teaspoon kosher salt. Decrease the heat to low and simmer until creamy and thick, stirring often, 45 to 60 minutes.
Heat a large cast-iron skillet over medium-high heat. Add the country ham and cook until crisped and golden brown on each side, about 5 minutes. Remove the ham and keep warm. Add the coffee and 1 cup water to the skillet, scraping up any brown bits with a wooden spoon. Simmer until the "gravy" has reduced to about 1/2 cup, 3 to 5 minutes.
Spoon the hot grits into warmed bowls. Top each with a pat of butter and black pepper to taste. Top with the ham and ladle over the gravy. Serve immediately.