Wrap the sweet onion and shallot in a pouch with olive oil, salt, and pepper, then sprinkle with sweet paprika. Bake in the oven at 375°F until they become soft.

Dice the sweet onion and chop a couple of cloves of garlic. In a pan, sauté the onions until they are soft, then add the chopped garlic along with salt and pepper.

Add a generous amount of tomato paste to the pan and caramelize it to enhance the flavor.

Deglaze the pan with a bit of chicken stock, scraping up the brown bits from the bottom.

Combine the roasted vegetables with the mixture in the pot and stir everything together.

Pour in another splash of chicken stock into the pot.

Use an immersion blender to blend the mixture until it reaches a smooth consistency.

Stir in heavy cream to add a touch of sweetness.

To prepare the aioli, mix together roasted garlic, Parmesan reggiano, basil, toasted pine nuts, lemon zest, lemon juice, vinegar, Dijon mustard, salt, and pepper, gradually adding oil to thicken the mixture.

Grate the extra sharp white cheese and slice the white American cheese.

Spread a thin layer of basil aioli on the bread, then layer on the cheese and add a bit more aioli before grilling.

In a pan, heat butter and place the sandwich sauce side down, cooking on medium-low heat.

After grilling, flip the sandwich and add more butter if necessary.

Serve the sandwich with a sprig of basil and a drizzle of heavy cream.