Preheat the oven to 400°F. Line a baking sheet with parchment paper or a Silpat.
In a small bowl, mix together the shredded and grated Parmesan cheese. Spoon a heaping tablespoon of the cheese onto the baking sheet and flatten it slightly. Repeat the scooping and flattening process with the remaining cheese, spacing the mounds at least 1 inch apart.
Bake the cheese for 5 to 7 minutes until golden around the edges. Remove the baking sheet from the oven and let the crisps cool completely on the baking sheet.
Once the crisps have cooled, assemble the sandwiches by toasting the bread, spreading two slices with mayonnaise and adding the bacon, lettuce, tomato and three crisps per sandwich. Top each sandwich with a second slice of bread and serve immediately.