Step 1

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.

Step 2

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.

Step 3

Preheat the oven to 400 degrees F.

Step 4

To a small bowl, add the olive oil, rosemary, thyme, garlic and salt and cracked black pepper to taste and whisk until blended.

Step 5

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.

Step 6

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.

Step 7

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.

Step 8

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.

Step 9

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.