Heat the oil in a medium saucepan over medium heat. Add the onion and cook, stirring, until tender, 3 to 5 minutes. Add the potato, cayenne, garlic, 1/2 teaspoon salt and a few grinds of pepper and cook, stirring, until the potatoes are well coated, 2 to 3 minutes. Pour in the broth and 1 cup of water and bring to a simmer.
Add the broccoli and simmer until the potatoes and broccoli are tender, about 15 minutes, adjusting the heat as needed to maintain a simmer. Stir in the kale and continue to cook until the kale is tender, about 5 minutes. Remove from the heat, and let cool at least 5 minutes. Carefully puree until smooth with a handheld immersion blender or in 2 to 3 batches in a blender (do not fill blender more than halfway full).
Reheat the soup if necessary. Off the heat, swirl in the butter until it is melted. Adjust the consistency with water and seasoning with salt and pepper as needed.
Serve hot sprinkled with croutons and grated Pecorino.
When blending hot liquid, first let it cool for 5 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.
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