Fill a large Dutch oven two-thirds full with oil and heat over medium-high heat to 360 degrees F.
Add 2 tablespoons oil to a large heavy-bottomed skillet over medium-high heat. Add the ham and cook just until browned, 4 to 5 minutes. Remove with a slotted spoon to a large bowl. Add the grits and cheese to the bowl and stir until combined. Using wet hands, scoop 1/4 cup of the grit mixture, roll it into a ball and place on a platter or large plate. Repeat until all the grit mixture is used.
Whisk together the flour, paprika and cayenne in a shallow bowl. In a separate shallow bowl, add the eggs. Add the panko to another separate shallow bowl. Dip each grit ball into the flour mixture and shake to remove any excess before dipping into the eggs and then into the panko, spinning to coat completely.
Add the grit balls to the Dutch oven in batches, frying until brown on all sides, about 5 minutes. Remove to a sheet pan lined with a wire rack and immediately sprinkle with salt.