Position two oven racks, one in the upper third and one in the lower third of the oven. Preheat the oven to 425°F.

In a Dutch oven over medium heat, melt the butter and cook until it smells nutty and the milk solids turn golden brown, about 3–5 minutes. Remove from the heat.

Add the whole tomatoes with their juices, sliced onion, smashed garlic, tomato paste, thyme sprigs, avocado oil, kosher salt, and black pepper. Stir until everything is well coated in the browned butter.

Roast the Dutch oven, uncovered, on the lower oven rack for 40–45 minutes, stirring once halfway through. The tomatoes should be bubbling, the onion should be tender, the garlic should be soft, and the tomato paste should look caramelized around the edges.

Transfer the Dutch oven to the stovetop over medium heat. Remove and discard the thyme sprigs. Pour in the stock, scraping up any browned bits from the bottom of the pot.

Add the Parmesan rind and simmer for 20 minutes.

Remove and discard the Parmesan rind. Blend the soup until completely smooth using a high-powered blender, or use an immersion blender directly in the pot.

Return the soup to the pot, if needed. Stir in the balsamic vinegar and heavy cream. Taste, then add brown sugar to balance the acidity. I usually use about 2 tablespoons. Add more salt and pepper as needed.

While the soup simmers, make the grilled cheese. Brush one side of each slice of sourdough with avocado oil. Place the bread oil-side down on a plate or cutting board.

Spread a thin layer of Boursin on the inside of each slice of bread. Top 4 slices evenly with the shredded white cheddar and Gruyère. Sprinkle with Parmesan, if using, then close the sandwiches.

Heat a large nonstick or cast-iron skillet over medium-low heat. Add a little extra oil if needed. Place the sandwiches in the skillet and cover with a lid.

Cook for 3–4 minutes, or until the bread is deeply golden and crisp. Carefully flip and cook for another 3–4 minutes, until the second side is golden and the cheese is fully melted. Reduce the heat if the bread is browning too quickly.

Let the grilled cheese rest for 1–2 minutes before slicing.

Serve the tomato bisque hot with a swirl of cream, freshly grated Parmesan, and cracked black pepper. Serve with the crispy Boursin grilled cheese for dipping.