In a small saucepan, combine 1/2 cup of the oil, carrot, celery, cauliflower, thyme, oregano, and water. Bring to a boil, lower the heat, and simmer, covered, until the vegetables are somewhat tender, about 6 minutes. Transfer the vegetable mixture to a small bowl, and stir in the olives, pimento, and salt. Set aside to cool.

When the olive-vegetable salad has cooled, stir in the parsley.

Slice the bread in half lengthwise. Using your hands, scoop out the inside of each bread half to make a shallow pocket. Brush the inside of the top half with the remaining 2 tablespoons oil. Fill the bottom bread half with the olive-vegetable salad and pack it down with your hands. Layer the cheese, mortadella, and soppresata over the salad (without letting any hang over the sides). Cover with the top half of bread and tightly wrap the sandwich with plastic wrap. Place the sandwich on a cutting board or baking sheet and weigh it down with a heavy object, such as a cast-iron skillet, for at least 1 hour or up to 3.

Remove the plastic wrap and cut the sandwich into fourths or sixths. Serve.