Combine the extra virgin olive oil, vinegar, honey, garlic, shallot, oregano, and basil in a large mixing bowl.
Add the vegetables and marinate for about half an hour.
Preheat the oven to 450ºF.
Heat a grill to medium-high heat.
Grill the vegetables, moving frequently to char and avoid flare ups, until just tender.
When cool enough to handle, slice the vegetables in half and toss to combine
To assemble: Stack on the bottom of each roll, 1/4 of the salami, then 1/4 of the grilled vegetables, then top with 1/4 of the Fontina cheese.
Bake the sandwiches until the cheese is melted and beginning to brown.
Top each with some of the onion, then tomato, then arugula, and place the top of the rolls on each.
Cut in half and serve hot.