For the soup: In a large saucepan or Dutch oven, heat the oil over medium heat. Add the shallots and season with a pinch of salt and pepper. Cook, stirring frequently, until tender, about 3 minutes. Add the garlic and cook until aromatic, about 30 seconds. Add the potatoes, chicken broth, and arugula. Bring to a boil. Reduce the heat, cover, and simmer until the potatoes are very tender, about 15 minutes. In batches, ladle the mixture into a blender or food processor and blend until smooth. Return the soup to the saucepan and whisk in the cheese, 3/4 teaspoon salt, and 1/4 teaspoon pepper. Let the soup cool to room temperature, about 30 minutes. Refrigerate for at least 1 hour until ready to serve. Before serving, adjust the seasoning with salt and pepper, to taste.
For the croutons: Place an oven rack in the center of the oven. Preheat the oven to 375 degrees F.
In a small bowl, toss together the bread, oil, and salt until coated. Spread in a single layer on a small baking sheet. Bake until golden, 10 to 12 minutes. Cool for 10 minutes.
To serve: Ladle the soup into small bowls. Garnish with croutons and cherry tomatoes.
Recipe courtesy of Giada De Laurentiis