Preheat the oven to 225 degrees F.
Prepare the dipping sauce by combining the egg yolks, lemon juice, garlic, and mustard in the bowl of a food processor. Process on high speed for 30 seconds. With the machine running, slowly add the olive oil and extra virgin olive oil and process until the mixture becomes a smooth emulsion. Add the salt and cayenne, and adjust seasoning to taste. Transfer to a decorative bowl and refrigerate until needed.
In a heavy 6-quart pot, heat the vegetable oil to 350 degrees F.
In a bowl, combine the flour and Essence. In batches, add the squid and shrimp and toss to coat. Shake to remove any excess.
In batches, add the squid and shrimp to the oil and cook until golden and crisp, 2 to 3 minutes. Remove with a slotted spoon and drain on paper towels. Transfer to a baking sheet and place in the oven to keep warm while cooking the remaining ingredients.
Toss the vegetables in batches in the seasoned flour and add to the oil, ensuring that the oil has returned to 350 degrees F before cooking each batch. Cook the vegetables until crisp and golden brown, 2 to 3 minutes. Remove and drain on paper towels, and place in the oven as needed while cooking the remaining vegetables. Sprinkle vegetables with grated cheese.
Serve the fritto misto hot on a large platter with the lemon-garlic dipping sauce.
*RAW EGG WARNING
The American Egg Board states: "There have been warnings against consuming raw or lightly cooked eggs on the grounds that the egg may be contaminated with Salmonella, a bacteria responsible for a type of foodborne illness.… Healthy people need to remember that there is a very small risk and treat eggs and other raw animal foods accordingly. Use only properly refrigerated, clean, sound-shelled, fresh, grade AA or A eggs. Avoid mixing yolks and whites with the shell…"
Yield: 4 to 6 servings