For the ratatouille: Fit a baking sheet with a rack. Line an 8 1/2-by-4-inch loaf pan with a sheet of aluminum foil that is as wide as the pan and about 12 inches long, making a cradle for easier removal of the ratatouille loaf.
Slice the eggplant lengthwise into 1-inch-thick planks. Slice each plank lengthwise in half again. Put the eggplant, zucchini and squash on the rack in a single layer. Generously salt both sides of the vegetables and let sit for 30 minutes to draw out the moisture, then pat dry with paper towels.
Preheat the oven to 400 degrees F.
To a small bowl, add the olive oil, rosemary, thyme, garlic and salt and cracked black pepper to taste and whisk until blended.
Assemble the vegetables in layers of each type in the loaf pan with the skins facing towards the walls of the pan: yellow squash, red pepper, eggplant, onion and zucchini, brushing each layer with the herb-oil mixture and sprinkling with 1 teaspoon of cheese. Press firmly to form a tight loaf while layering. Repeat until the vegetables reach about 1/2 inch from the top of the loaf pan. Top with the tomato rounds and sprinkle with the remainder of the Parmesan and some salt.
Put the loaf pan on a rimmed baking sheet and bake uncovered until the vegetables are very soft and the tomatoes are soft and caramelized, about 1 hour 15 minutes. Let cool completely, at least 4 hours, or refrigerate overnight.
When ready to serve, toss the arugula with a little (about 1 to 2 tablespoons) of the Champagne Vinaigrette to lightly dress. Remove the ratatouille from the pan by pulling it up with the ends of the foil cradle. Slice the ratatouille crosswise into 4 even pieces.
For the sandwich build: Split the rolls in half lengthwise, brush both sides with butter, toast on a griddle and rub with the cut side of the garlic. Spread some of the butter on the cut side of the rolls and sprinkle with the flaky salt. Add a slice of the ratatouille and top with the arugula. Cut each sandwich in half on the bias.
In a medium bowl, whisk together the vinegar, parsley, mustard, oregano, sugar, thyme and garlic. While whisking, slowly add the olive oil. Season with salt and pepper and you're ready to rock.