In a food processor bowl, add deli ham and next 6 ingredients. Pulse until finely ground and thick. Spoon mixture into a 1 quart re-sealable plastic bag. Cut off the tip from one bottom corner and squeeze salad mixture towards the hole.
Layout 8 bread slices on a large baking sheet. Pipe mixture onto bread slices in the shape of a tombstone, leaving edges unfilled to allow for a narrow tombstone shape. Top each with 1 slice of Sargento® Mild Cheddar Cheese. Place remaining slices of bread over cheese. Cover pan of sandwiches with plastic wrap. Refrigerate for 1 hour.
Remove sandwiches from refrigerator and unwrap. Place 1 sandwich at a time onto a cutting board and cut off sides and bottom of bread to create a clean-cut edge, making the sandwich resemble a tombstone. Discard cut sides and bottom of of sandwiches.
Spoon ketchup or hot sauce into a 1 quart re-sealable plastic bag. Cut a small tip off one end and squeeze ketchup toward the opening.
Write the letters RIP in ketchup onto the top of the sandwiches. Carefully transfer sandwiches to a platter topped with rye chip crumbs. Serve.