1 cup canola oil
8 slices prosciutto
16 to 24 fresh sage leaves, stemmed
Kosher salt
1/2 cup mayonnaise
1/4 cup red wine vinegar
1 to 1 1/2 pounds whole skinless, boneless chicken breast, cut into 1/4-inch-thick pieces and pounded very thin
1 heaping tablespoon dried oregano
3 eggs, lightly beaten
4 cups plain breadcrumbs
4 large crusty, seeded rolls, split
2 romaine hearts, cut into 1/2-inch rounds
1/4 cup grated Parmesan