For the pesto: Pulse together the basil, walnuts, walnut oil and garlic in a food processor until combined. Add the Parmesan and olive oil and pulse again.
For the sandwiches: Cut the pumpernickel into 1/2-inch slices, and then into coffin shapes. On each slice of bread, spread a layer of mayo, a layer of pesto, a sprinkling of almonds, alfalfa sprouts, a layer of apples, and shaved Cheddar.
Place another coffin-shaped bread slice on top and repeat for another layer. Top off with a final coffin slice.