How to Make It
In a large bowl, whisk the flour with the sugar, salt, baking powder and baking soda. Pinch the butter into the dry ingredients to form pea-size clumps. Stir in the buttermilk, scallions and the beaten egg just until a dry, shaggy dough forms. Turn the dough out onto a lightly floured work surface and knead gently just until it comes together. Pat the dough out to a 1/2 -inch-thick rectangle. Using a 4-inch round cookie cutter, stamp out 6 biscuits. Gently press the scraps together and stamp out 2 more biscuits. Arrange the biscuits on a parchment paper–lined baking sheet and freeze for 30 minutes.
Preheat the oven to 375°. Bake the biscuits for about 20 minutes, until golden brown. Let cool slightly, then split in half, laying the halves cut side up on the baking sheet. Cover each biscuit half with 1 cheddar slice and 1 ham slice. Bake until the cheese is melted, about 8 minutes.
In a large nonstick skillet, heat the canola oil. Crack 4 eggs into the skillet and cook sunny side up over moderately high heat until the whites are set and the yolks are runny, 2 to 3 minutes. Set the fried eggs on 4 of the biscuit bottoms and keep warm while you fry the remaining eggs. Close the sandwiches and serve immediately.