Heat the oven to 425°F (220°C). Line a baking sheet with parchment paper.
Load the pan. Cut 2½ lb ripe tomatoes into 6 wedges each and spread them out in one layer. Tuck the cloves from 1 head garlic between them. Not grilling the pepper? Cut it up and add it to the pan now. Drizzle with 4 Tbsp olive oil, keeping 2 Tbsp back if you caramelize the onions in a skillet. Don't crowd the pan - use two pans for a big batch.
On the pan: cut 2 yellow onions in half, then each half into 4-6 pieces, and scatter them over the tomatoes. In a skillet: dice them and cook in 2 Tbsp of the olive oil in a skillet over medium heat while the tomatoes roast, stirring now and then, until deep golden (Note 2).
Roast 20-25 minutes, until the tomatoes soften and char on the tops. Pull out any garlic cloves or onion pieces that turn dark brown.
On the grill: grill 1 bell pepper (or 5 mini sweet peppers) whole over medium heat, turning, until soft with dark grill marks. In the oven: it roasts on the pan with the tomatoes from step 2 (Note 1).
Load the blender. Add the roasted tomatoes and garlic with all the pan juices, the pepper (stem and seeds removed), the onions (with their oil, if made in the skillet), ½ tsp kosher salt, 1 tsp black pepper, ⅛ tsp cayenne pepper, ¼ tsp Italian seasoning (or 4 fresh basil leaves) and about a third of the sugar (Note 5). Fill the blender no more than halfway, in two batches if needed.
Blend until smooth on high, about 2 minutes, with the center cap out and a folded towel held over the opening so steam can escape.
Add broth and cream. With the blender on low, slowly pour in 1½ cups chicken broth through the lid opening, then ¼ cup heavy cream. Add a little more broth for a thinner soup (Note 3).
Blend in the Parmesan. Add ½ cup Parmigiano-Reggiano and blend on low until it melts into the soup.
Taste and adjust. Blend in the rest of the sugar a little at a time if the soup tastes sour (Note 5). Taste before adding more salt - the Parmesan is salty. Serve hot (Note 6).
1. Peppers: any color works. I used yellow and orange mini sweet peppers; a red bell pepper turns the soup a deeper red. Grilling adds a light smoky taste, and roasting it on the pan with the tomatoes is quicker.
2. Onion, two ways: roast it on the pan with the tomatoes, or dice it and caramelize it in a skillet with 2 Tbsp of the olive oil while the tomatoes roast. Either way, pull out or avoid dark, burnt bits, which turn the soup bitter. Onion from the skillet goes into the blender with its oil.
3. Broth: hot broth keeps the soup hot as it blends. Hot water with 1 Tbsp Better Than Bouillon works too, and vegetable broth works too. For a vegetarian soup, use vegetable broth and a vegetarian Parmesan-style cheese, since real Parmigiano-Reggiano is made with animal rennet.
4. Cream: hot cream blends straight into the hot soup. Leave it out for a lighter soup, or use more broth instead.
5. Sugar: it balances the tomatoes' acidity. Ripe summer tomatoes may need only a third of it; off-season tomatoes usually need it all.
6. Storage: fridge up to 4 days, freezer up to 3 months. For the smoothest soup, freeze it before adding the cream and stir the hot cream in after reheating.
7. Make ahead: roast the vegetables up to 3 days early, keep them covered in the fridge, then blend with hot broth and cream and warm the soup in a pot.
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