Make the soup: Preheat the oven to 400 degrees F. Heat the olive oil in a Dutch oven or heavy-bottomed pot over medium-high heat. Add the onion, garlic and 1 teaspoon salt and cook, stirring occasionally, until the onion is soft, about 5 minutes. Add the chicken broth, 3 cups water, the potato and the parmesan rind. Season the chicken with salt and pepper and add to the pot. Bring to a boil, then reduce the heat to medium and simmer until the chicken is cooked through, about 15 minutes. Remove the chicken to a plate using tongs and let cool.
Meanwhile, make the croutons: Combine the mayonnaise, grated parmesan, lemon juice, anchovies, hot sauce, garlic, a few grinds of pepper and 1 tablespoon water in a blender or food processor; puree until smooth.
Cut the baguette in half lengthwise, then slice into 1/2-inch-thick pieces. Arrange on a baking sheet and brush with the mayonnaise mixture, reserving 2 tablespoons for topping. Sprinkle with the parsley, lemon zest and more parmesan; bake until golden, about 10 minutes.
Add the lettuce and heavy cream to the pot and cook until the lettuce is tender, 8 to 10 minutes. Discard the parmesan rind. Transfer the soup to a blender and puree until smooth. Return to the pot and season with salt and pepper. Shred the chicken and add it to the soup; heat through over medium heat. Divide among bowls, drizzle with the reserved mayonnaise mixture and top with the croutons and more pepper.
Photograph by Johnny Miller
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.