Heat the olive oil in a medium pot over medium heat. Add the onion and sweat for 3 minutes to soften. Stir in the garlic and let cook for another 2 minutes. Add all the remaining jam ingredients, except the parsley. Stir in 1/4 cup of water along with the onions and bring to a simmer. Turn the heat to low and let simmer for 25 to 30 minutes until most of the liquid has reduced and the tomatoes are soft. Remove from the heat and stir in the parsley.
Pound out each piece of chicken until it's about 1/4-inch thick. Slice each breast into 6 pieces. Season generously with salt and pepper, to taste
Heat the olive oil in a large skillet over medium-high heat. Working in batches, fry the pancetta until brown and crispy. Remove to a plate lined with a paper towel.
In the same pan that the pancetta was cooked in, fry the chicken breast pieces over medium-high heat, in batches, for 2 minutes. Flip and cook for 2 more minutes. Top each piece of chicken with a slice of mozzarella cheese, cover with a lid and cook until the cheese is melted.
To assemble, put the chicken breast pieces onto the bottom half of each roll. Top with a slice of pancetta, then about 1 tablespoon of the tomato jam and a basil leaf. Cover with the top half of the roll and serve.