Preheat the oven to 200°C (400°F).
On a large baking sheet, arrange the cherry tomatoes, red bell pepper, diced potatoes, halved onion, and 2 heads of garlic (with the tops cut off).
Add the 2 sprigs of rosemary on top of the vegetables.
Season generously with salt and pepper, and drizzle with 3 tbsp of olive oil.
Roast the vegetables in the oven for 45 minutes, or until they are tender and caramelized. The tomatoes should be blistered, and the pepper should be slightly charred.
After roasting, transfer the vegetables (discard the rosemary sprigs) into a large bowl.
Squeeze the roasted garlic cloves out of their skins into the bowl with the vegetables.
Add 300g of water (or vegetable stock), chili flakes, and 3 tbsp of fresh parsley.
Use an immersion blender (hand blender) to blend everything in the bowl until smooth and creamy. Adjust the consistency by adding more water if needed.
Pour the blended mixture into a pot and heat over medium heat.
Stir in 200g of heavy cream and gently bring the soup to a simmer over medium heat.
Taste and adjust seasoning with additional salt, pepper, or chili flakes if needed.
Let the soup simmer for about 5 minutes, stirring occasionally.
Slice the bun in half and place 50g of Emmental cheese between the slices.
Grill the sandwich in a hot pan until the cheese is melted and the bread is golden brown, about 3-4 minutes per side.
Ladle the soup into bowls and garnish with a drizzle of olive oil, a sprinkle of chili flakes, and fresh parsley.
Serve with the warm, crispy Emmental cheese sandwich on the side.