In a medium bowl, mix together the ricotta, Parmigiano-Reggiano, oil and rosemary with a rubber spatula. Set aside.
Lay two pieces of the prosciutto cotto on the board in front of you lengthwise, overlapping by half and patching holes where needed. Spread about 2 tablespoons of the cheese mixture evenly over the ham. Sprinkle the side closest to you with 2 tablespoons chopped giardiniera and press gently into the cheese. Roll the ham into a pinwheel starting with the edge closest to you and rolling away from you. Repeat for the remaining prosciutto cotto and filling. Wrap each roll tightly in plastic wrap and store in the refrigerator for at least 4 hours to firm up.
To serve, unwrap the rolls and, using a sharp knife, slice into 3/4-inch pieces.
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