Preheat a grill for cooking over medium-high heat. Lightly oil the grill grates.
Drizzle the cut-sides of the tomato halves with olive oil and sprinkle with salt and pepper. Place them on the grill cut-side down until they have very dark grill marks, about 8 minutes. Gently flip and grill on the skin-side to char, about 2 minutes. Lightly drizzle some olive oil on a sheet of aluminum foil. Arrange the onion rounds in one layer on the foil. Drizzle with a little more olive oil and sprinkle with salt and pepper. Place the foil sheet on the grill grates. Grill until the onions develop dark char on both sides, about 5 minutes per side. Turn the grill down to medium for the grilled cheese toast.
Heat the 1 tablespoon butter in a large pot or Dutch oven over medium heat. Add the grilled onions and cook until soft and translucent, about 3 minutes. Add the garlic and thyme; stir for about 1 minute. Add the tomatoes and all their juices and mash down the tomatoes with a wooden spoon to break them up a bit. Add 2 cups water, 2 teaspoons salt and some more black pepper. Bring the soup to a boil, then turn down to a simmer and cook, about 15 minutes. Take the soup off the heat, and stir in the basil leaves. Very carefully puree the soup in a blender or with a handheld stick blender. When ready to serve, bring the soup back to a simmer, and add the heavy cream. Taste for seasoning. Add more salt, if necessary. If the tomatoes are too acidic, add a little sugar to taste.
Meanwhile, make the grilled cheese toast: brush the melted butter on both sides of each bread slice. Place the slices on the grill; cook until nicely toasted on the underside, 3 to 5 minutes. Flip the bread and top each with 2 slices of cheese. Cover and cook to melt the cheese and toast the other side, 2 to 3 minutes. Grind a little pepper on each cheese toast. Serve immediately with the soup.
When blending hot liquids, first let cool for 5 minutes or so, then transfer to a blender, filling only halfway. Put the lid on, leaving one corner open; this will prevent the vacuum effect that causes heat explosions. Cover the lid with a kitchen towel to catch splatters and pulse until smooth.
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