Portuguese Kale and Potato Soup

Heat 1 tablespoon oil in a 5-quart heavy pot over medium-high heat until it shimmers, then brown sausage, stirring often, 2 to 3 minutes. Transfer with a slotted spoon to a bowl. Add 2 tablespoon oil to fat in pot and cook onion and garlic with 1/4 teaspoon each of salt and pepper over medium heat, stirring often, until browned, 7 to 8 minutes.

Add potatoes, water, and 1 teaspoon salt and simmer, covered, until potatoes are very tender, 15 to 20 minutes. Mash some potatoes into soup to thicken, then add kale and simmer, uncovered, until tender, about 5 minutes. Stir in sausage and cook until just heated through, 1 to 2 minutes. Drizzle with remaining tablespoon oil and season with salt and pepper.