Preheat your oven to 425°F and drizzle half the olive oil onto a sheet tray, then add the cherry tomatoes, onion, and garlic and drizzle the rest of the oil on top with a generous sprinkle of salt and pepper.
Roast everything for 35 to 45 minutes until the tomatoes burst and the edges caramelize, then let it cool for about 5 minutes.
Transfer the roasted veggies and all the pan juices to a blender and blend on high until smooth, then taste and add a pinch of sugar or more salt if it needs balance.
While blending, bring 2 cups of water to a boil, pour half onto the sheet tray to loosen the browned bits, and blend that into the soup a little at a time until you get the consistency you like.
Add the cream and blend for 20 more seconds, adding extra hot water ¼ cup at a time if the soup still feels too thick.
Spread 1 tablespoon of mayo and 1 tablespoon of butter on one side of each bread slice, then press in ¼ cup of Parmesan onto the mayo sides and flip the slices over.
Layer the cheddar, Gouda, and Fontina onto one slice of bread, top with the other slice (Parmesan side facing out), and press together gently.
Heat 1 tablespoon of butter in a skillet over medium heat, add the sandwich, and cook for 3 to 4 minutes per side until golden brown, covering with a lid near the end to help melt the cheese.
Transfer the sandwich out of the pan, reduce the heat, sprinkle the rest of the Parmesan straight into the skillet, then set the sandwich back on top and cook for 1 to 2 more minutes until the bottom is crispy.
Let the sandwich cool for a few minutes, slice it in half, and serve hot with a big bowl of tomato soup and a sprinkle of parsley.