Heat the drippings or oil in a medium saucepan over medium heat. Add the onion and cook, stirring, until tender, 3 to 5 minutes. Add the potato, bay leaf, garlic, rosemary, 1/2 teaspoon salt and a few grinds of pepper and cook, stirring, until the potatoes are well coated, 2 to 3 minutes more. Pour in the broth and bring to a simmer.
Simmer until the potatoes are tender, about 15 minutes, adjusting the heat as needed to maintain a simmer. Stir in the escarole and cook 5 minutes more. Remove from the heat, discard the bay leaf and rosemary 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 the blender more than halfway full per batch).
Reheat the soup if necessary, adjusting the consistency and seasoning with additional water, salt and pepper.
Serve hot sprinkled with grated pecorino.
Disclaimer: 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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