Combine sun-dried tomatoes and 1/4 cup of their oil, capers, parsley, shallot, garlic, and vinegar in a small bowl and stir with a fork. Season to taste with salt and pepper. Heat oil in a large skillet over medium-high heat until shimmering. Add sausage and cook, breaking it up with a wooden spoon or a potato masher, until fully cooked and just starting to brown, about 8 minutes. Transfer to a bowl, leaving fat in skillet. Add onions, fennel, and peppers to skillet and cook over medium-high heat, stirring occasionally, until fully softened and just starting to brown, about 12 minutes. Season to taste with salt and pepper. Transfer to a plate lined with a double layer of paper towels and press with more paper towels to drain completely. Spread half of relish on bottom of sandwich and top with half of pepper/fennel mixture. Add 1/3 of cheese, then 1/2 of sausage, another 1/3 of cheese, remaining sausage, remaining cheese, remaining peppers and onions, and remaining relish. Close sandwich. Wrap sandwich tightly in aluminum foil or butcher's paper. Adjust oven rack to center position and preheat oven to 375°F. Place in oven until hot and cheese has melted, about 15 minutes. Serve immediately.