1 pound (450g) store-bought or homemade fresh lasagna noodles (if homemade, follow the recipe and instructions here through step 10), cut into 6- by 12-inch rectangles
Vegetable oil, for greasing pan and pasta
8 ounces (227g) fontina, rinds trimmed and cut into 1/4-inch pieces
1 cup finely grated Parmigiano-Reggiano (4 ounces; 112g), divided, plus more for garnish
8 ounces (227g) prosciutto cotto, torn into into 2-inch pieces
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