Warm the country ham in a dry skillet over medium heat for 45 seconds per side, then transfer it to a plate.
Melt 1 tablespoon butter in the skillet and fry the eggs over-medium with black pepper until they reach 160°F.
Transfer the eggs to the plate with the ham.
Add the remaining butter, coffee and brown sugar to the skillet and stir for 30 seconds until glossy.
Brush the red-eye butter lightly over the cut sides of the biscuits.
Fill each biscuit with country ham and one egg, keeping the yolk centered.