Heat the olive oil in a large pot over medium-high heat. Add the onion and celery and cook, stirring, until softened, about 5 minutes. Add the chicken broth, half-and-half, both potatoes, bay leaf, 2 cups water, 1 teaspoon salt and 1/4 teaspoon pepper and bring to a boil.
Reduce the heat to medium low and simmer until the potatoes are tender, about 10 minutes. Meanwhile, put the broccoli in a microwave-safe bowl; add 3 tablespoons water and season with salt. Cover with plastic wrap and microwave until crisp-tender, about 4 minutes; drain.
When the potatoes are tender, remove the bay leaf from the soup and transfer the soup to a blender. Puree until smooth, leaving the filler cap slightly open to let steam escape (or puree the soup in the pot with an immersion blender). Return the soup to the pot and thin with water, if necessary. Return to a simmer over medium-low heat; stir in the broccoli and season with salt and pepper.
Add the cheese and stir until melted. Ladle the soup into bowls and top with croutons, if desired.
Photograph by Antonis Achilleos
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.