Place the sliced country ham and the milk in a resealable plastic bag and let it sit overnight in the refrigerator.
Remove the ham and wipe dry with paper towels.
Heat the oil in a skillet over medium heat.
Add the ham and cook briefly until the fat is cooked through and the ham has reached a surface temperature of 160 degrees (or follow the package directions for cooking).
Remove the ham from the skillet and set aside. Keep the skillet over medium-high heat.
Mix the coffee, water and cornstarch together and pour into the pan.
Cook while stirring to dislodge all the drippings from the bottom of the pan. Reduce the heat and let the gravy simmer briefly to finish cooking.
Serve over plain or cheese grits, split biscuits, or over the cooked ham.
