Step 1Bring 2 cups water to a boil in a medium saucepan over high heat, then add salt and whisk in polenta in a steady stream. Lower heat to a simmer and cook for 10 minutes, stirring occasionally with a whisk to avoid lumps and until the polenta is thickened. Add the olive oil and Parmigiano-Reggiano and whisk to incorporate. Remove the pan from heat.
Step 2Pour the polenta onto a parchment-lined baking sheet with sides. Using a spatula, smooth the polenta to an even height of one-half inch. Refrigerate, uncovered, until set, about 45 minutes. Remove the chilled polenta from the pan and cut into one-half-inch cubes.
Step 3Fill a large saute pan with enough canola oil to cover the bottom by 1 1/2 inches. Heat the oil over high heat until a thermometer inserted reads 375 degrees. Adjust the heat to maintain this temperature. Fry a few croutons at a time, leaving enough space so they do not stick together. Be careful that the croutons do not stick to the bottom of the pan and burn. Fry the croutons until golden, about 2 minutes, then remove and set them on paper towels to drain.
Step 1Heat 1 tablespoon of the oil in a small saute pan set over medium heat. Add the anchovies and garlic and cook until the garlic is aromatic, less than 1 minute. Remove the pan from the heat.
Step 2In a blender or food processor, blend the egg yolks. Slowly drizzle in the remaining olive oil. When all is well incorporated, add the garlic and anchovies, the cheeses and mustard and blend well. Lastly add the lemon juice. Adjust the seasoning. Refrigerate until ready to use.
Step 1Remove the outer leaves from the lettuce heads. Tear the remaining leaves into bite-sized pieces and wash in cold water. Drain and spin dry. Place the lettuce in a large bowl; you should have about 8 cups salad.
Step 2Dress the salad with the dressing, and toss to evenly coat. Divide among four chilled plates. Use a vegetable peeler to shave a few strips of Parmigiano-Reggiano on top of each salad. Place one crouton on top of the cheese in the center of each salad, and evenly distribute the remainder of the croutons among the salads. Serve immediately.