For the corncakes: Mix together the self-rising flour, cornmeal, granulated sugar and salt in a medium bowl. Add the chicken stock, buttermilk, melted butter, eggs and yolks and mix until combined; do not overmix.
Oil a griddle and heat over medium heat. When hot, add 1/2 cup of the batter to the pan to create a small cake; repeat to fill the pan (about 2). Let the cakes cook until browned and the tops have begun to set slightly, 2 to 3 minutes. Flip the cakes and cook until golden brown on the other side and cooked through, another 1 to 2 minutes. Remove the cakes to a paper towel-lined tray to rest. Repeat with the remaining batter to make a total of 4 to 6 cakes.
For the Red-Eye Gravy: Heat 1 tablespoon butter in a large saute pan over medium-high heat until melted, then add the ham. Brown the ham on each side. Remove the ham and deglaze the pan with the coffee. Add the chicken stock and brown sugar and increase the heat to high. Allow the mixture to reduce by at least half, about 5 minutes. Remove from the heat. Add the 2 tablespoons ice-cold butter a little at a time, whisking constantly until all the butter has been added.
For the Country Ham Scramble: Whisk the eggs in a medium bowl. Add the cream and whisk well, then add the Cheddar. Heat a large nonstick saute pan over medium-low heat. Add the egg mixture and scramble until soft.
To Plate: place a hot corncake on a plate, top with a quarter of the eggs, a sprinkle of ham and some red-eye gravy. Garnish with the green onions.