Place the butter in a large, deep pot over medium-high heat. When the butter is melted, add the shallot and broccoli florets, sprinkle with salt and pepper, and cook until the veggies start to soften, 5 to 8 minutes.
Add the water and bring to a boil. Then, lower the heat so the soup bubbles gently and cook, stirring occasionally, until slightly thickened and broccoli becomes fully tender, about 10 to 15 minutes longer.
Using an immersion blender, puree the soup with the chopped basil leaves in the pan; or let the mixture cool slightly and use a blender to process until smooth. Return the soup to the pan; taste and adjust the seasoning. Serve with a drizzle of olive oil.