To prepare the basil essence, in a blender, combine the olive oil and one-fourth of the basil leaves and puree. Continue adding the basil leaves until all of them are pureed.

Transfer the basil mixture to a nonstick sauté pan and set over medium heat. Cook, stirring continually, until the mixture appears to be completely separated, about 7 minutes. Strain the mixture through a paper coffee filter into a bowl and set aside until ready to use.

To make the tomato soup, prepare a smoker according to the manufacturer's instructions.

In a large bowl, toss the tomatoes with 2 tablespoons of the olive oil and season with salt and pepper. Smoke for 25 to 30 minutes. 

Preheat the oven to 425°F.

Place the tomatoes on a baking sheet and roast in the oven until the skins are blistered and the tomatoes have released their liquid.

Meanwhile, in a large saucepan over medium-low heat, warm the remaining 2 tablespoons olive oil. Add the onions and garlic and cook, stirring occasionally, until the onions are translucent. Add the roasted tomatoes (crush them in the pan) and all of the tomato liquid, the stock, oregano, thyme and bay leaves. Simmer, stirring occasionally, for 1 hour. Remove and discard the bay leaves.

Using an immersion blender, puree the mixture until smooth. Add the cream and bring the soup to a boil. Reduce to a slow simmer and cook until the soup reduces slightly. Pass the soup through a fine-mesh sieve and return it to the pot. Return the soup to a slow simmer and cook until light and silky in texture. 

When the soup is almost done, make the panini (recipe below).

To make the panini, preheat a panini press according to the manufacturer's instructions.

In a bowl, stir together the mayonnaise, thyme and parsley, and season with salt and pepper. 

Spread butter on one side of the bread slices. Place the slices, buttered side down, on a clean work surface. Spread 1 tablespoon of the mayonnaise mixture on each bread slice. Top half of the bread slices with 2 cheese slices and 2 tablespoons caramelized onions. Cover each sandwich with one of the remaining bread slices, buttered side up.

Working in batches, cook the sandwiches on the preheated panini press until warm and melted in the center. 

To serve, ladle the soup into bowls. Drizzle the basil essence over the top and serve with the panini.