Chef Marco Canora prefers to use a food mill for this savory recipe from his "Salt to Taste" cookbook because it removes the skins from the peppers, which can sometimes be bitter, and doesn't aerate the soup.
Heat soffritto base in a large pot over medium-high heat. Add peppers and cook, stirring to coat. Continue cooking until peppers are softened, about 3 minutes. Add potatoes and stir to combine.
Add bay leaves and enough water to cover the vegetable mixture by about 1 inch (about 9 cups); season with salt and pepper and bring to a boil. Reduce heat to a simmer and cook until pepper and potatoes are almost soft, about 30 minutes.
Crush the thyme with the flat side of a knife and add it to the pot. Continue to simmer until peppers and potatoes are completely soft, about 15 minutes more.
Remove bay leaves and thyme; discard. Working in batches, pass soup through a food mill fitted with a fine disk. Whisk in 1/2 cup olive oil, if desired; season with salt and pepper. If soup seems too thick, add a little water to thin out. Serve warm, drizzled with olive oil and garnished with croutons and thyme leaves.