Allow the pizza dough to sit at room temperature for at least 30 minutes.
Combine 1/4 cup of the Parmesan, lemon juice, vinegar, egg yolk, anchovies, garlic, 1/4 teaspoon salt and a few grinds of pepper in a medium bowl. Slowly drizzle in the olive oil while whisking until a smooth vinaigrette forms. Cover and refrigerate until ready to use.
Put the sun-dried tomatoes on a small plate and microwave, in 30-second increments, until they begin to harden but are still red and slightly pliable, 1 to 2 minutes (they may begin to brown and taste bitter if cooked too long). Let them sit until cool and completely hardened, about 2 minutes. Break up the tomatoes into small pieces, transfer them to a spice grinder and process to a powder (discard any pieces that won't grind). Add the remaining 1/4 cup Parmesan, oregano, garlic salt and 1/4 teaspoon salt. Process until just finely ground. Transfer to a large bowl.
Fill a medium pot with 1 inch of vegetable oil and bring to 350 degrees F over medium heat. Evenly divide the pizza dough into 4 balls. On a lightly-floured surface, roll and stretch each ball into a 4 1/2-inch round and cut into 4 wedges. In batches, carefully drop wedges into the oil and fry until puffed and golden, turning as needed, 2 to 3 minutes. Transfer the fried wedges to the bowl of pizza-spice mix and toss to coat. Remove and set aside. Fry and coat the remaining dough.
Put the romaine in another large bowl, add the Caesar dressing, toss to coat and season with salt and pepper. Divide among four serving plates. Top each with pizza croutons and sprinkle with any remaining pizza spice mix.
Raw Egg WarningFood Network Kitchens suggest caution in consuming raw and lightly cooked eggs due to the risk of Salmonella infection or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly refrigerated, clean, grade A or AA eggs with intact shells and avoid contact between the yolks or whites and the shell. For recipes that call for eggs that are raw or undercooked when the dish is served, use shell eggs that have been treated to destroy Salmonella, by pasteurization or another approved method.