Heat oven to 425 degrees F. Coat jumbo-muffin pan with 6 indentations with nonstick cooking spray.
Cut cheese in half; shred half and cut other half into 12 thin slices.
In bowl, whisk flour, cornmeal, baking powder, soda, salt, cayenne. Cut in shortening with pastry blender until very fine pieces. Stir in shredded cheese, then buttermilk. Gather dough together; transfer to floured surface. Knead twice.
Pat dough into 8 x 6-inch rectangle. Cut into 6 equal pieces; shape each into round, 3 inches in diameter. Transfer to ungreased baking sheet.
Fill 13 x 9 x 2-inch baking pan with 1 inch warm water. Crack 1 egg into each prepared muffin cup, discarding shells. Sprinkle eggs with salt and pepper. Place muffin pan in baking pan. Transfer to oven along with biscuit-filled baking sheet.
Bake biscuits and eggs in 425 degree F oven 20 minutes, until biscuits are golden brown and center of eggs are set. Transfer biscuits to rack; remove muffin pan from oven. Leave oven on.
With serrated knife, cut biscuits in half horizontally. Using a thin spatula, remove egg from each muffin pan and place on a biscuit bottom. Spread each with 1 tablespoon salsa. Top each with 2 cheese slices and 1 slice ham. Cover with matching biscuit top; return to baking sheet.
Heat sandwiches in 425 degree F oven until cheese is melted, 2 minutes. Serve warm with salsa, if desired. Makes 6 sandwiches.