Preheat an oven to 400F.
In a small bowl, combine the Parmigiano and the flour.
On a large nonstick baking sheet, place about 2 tablespoons of the grated cheese in circles about 4 inches across. Place the baking sheet into the oven until they start to smell nutty and the cheese is melted and golden brown, about 10 minutes.
Once the cheese is golden, remove the tray from the oven. Working quickly, lift up the cheese rounds with a small knife or spatula and press them into the cups of an egg carton with your fingers to shape them into small bowls. Let cool.
While the cheese bowls are cooling, combine the lettuce, anchovy paste, lemon juice and Parmigiano in a mixing bowl. Season with salt, freshly ground black pepper and a drizzle of EVOO, and toss to combine.
When the cheese cups are completely cooled, arrange them on a serving platter and fill each one with a small scoop of the Caesar salad. Serve immediately.