Warm the olive oil in a large pot over medium heat. Brown the sliced sausage until lightly crisp. Transfer to a plate.
Soften the onion in the rendered fat, about 5 minutes. Stir in the garlic and cook 30 seconds until fragrant.
Add potatoes, broth, and water. Season with salt, black pepper, and red pepper flakes. Bring to a boil, then simmer until potatoes are tender, 12–15 minutes.
Lightly mash the potatoes in the pot to thicken the soup, leaving some chunks for texture.
Stir in the kale and simmer until wilted and tender, 5–7 minutes.
Return the sausage to the pot and warm through. Taste and adjust seasoning. Ladle into bowls, drizzle with olive oil, and garnish with parsley.