Cook meatballs according to package instructions.
While the meatballs are cooking, chop the giardiniera and reserve.
Combine the tomatoes, shallots, tarragon, vinegar, oil, and season with salt and pepper, and reserve.
Combine the pesto and mayonnaise and reserve.
When the meatballs are done, remove from the oven, place on a plate lined with a paper towel, and allow to rest.
Turn the oven on to broil, and when hot, open the rolls and toast on the middle rack on a sheet pan.
Remove the rolls from the oven, and the tops from the pan.
Divide the meatballs evenly between each of the bottoms of the rolls, and evenly divide the cheese on top of the meatballs.
Melt the cheese under the broiler and remove and reserve.
Dress the top of the rolls with the pesto mayonnaise, evenly divide the tomatoes on top of the melted cheese, top the tomatoes with the giardiniera, and top with the dressed rolls.