Preheat a grill to medium high and lightly oil the grates. Butterfly the pork: Slice almost in half lengthwise, stopping about 1/2 inch from cutting through; open like a book and flatten with your hands. Cover with plastic wrap and pound to about 1/2 inch thick with the flat side of a meat mallet or a heavy skillet.
Rub the pork with the grated garlic and season with salt and pepper. Arrange the red peppers on top of the pork in a single layer, then the basil and mozzarella, leaving a 1-inch border on one long side. Roll up from the opposite long side into a tight log and tie with kitchen twine in three or four places. Rub with 1 tablespoon olive oil and season well with salt and pepper.
Grill the pork, covered and turning occasionally, until a thermometer inserted into the meat (not the filling) registers 145 degrees F, 12 to 14 minutes. Let rest about 5 minutes, then slice.
Meanwhile, strip the kale leaves from the stems and tear into pieces. Heat the remaining 3 tablespoons olive oil in a large pot or Dutch oven over medium-high heat. Add the kale, smashed garlic and a big pinch each of salt and pepper. Cook, tossing, until the kale is tender, about 5 minutes. Roughly chop the giardiniera and stir into the kale until heated through, 30 seconds to 1 minute. Divide among plates and serve with the pork.