Score the pork belly several times on fatty side. Season with salt and pepper, to taste.
Coat a large Dutch oven, over medium heat, with oil.
Add the pork belly, 1 piece at a time, scored side down, to the pot. Sear slowly to render the fat on all sides for 3 to 4 minutes.
Remove from pan and set aside.
Drain off the excess fat leaving 3 to 4 tablespoons and add the garlic. Brown the garlic, then add the onions and cook until slightly colored, about 2 to 3 minutes.
Stir in the brown sugar and balsamic and cook until syrupy, about 1 to 2 minutes.
Add the Dijon and whisk until smooth. Stir in the rosemary, mushrooms, peppers and broths and bring to a boil. Return the pork belly to the pan, cover and put in the oven. Braise until fork tender, about 1 1/2 to 2 hours.
Remove from oven and transfer the pork belly to a platter.
Mix all the ingredients with a whisk in a small bowl and set aside.
Spread pesto mayonnaise liberally on each side of the toasted rolls. Put a piece of pork belly and several slices of smoked mozzarella on the bottom halves. Arrange the bottom halves of assembled sandwich on a sheet pan and put back in the oven until cheese melts and browns, about 1 to 2 minutes.
Remove from the oven, cover with the roll tops and serve.
