Heat oil in a large pot or Dutch oven over medium.
Add onion, ¼ teaspoon salt and ¼ teaspoon red pepper flakes; cook until softened and translucent, about 5 minutes.
Add sausage and cook, crushing with a wooden spoon, until cooked through and browned, 6–8 minutes. Drain all but 2 tablespoons drippings from pot.
Meanwhile, quarter fennel bulbs lengthwise, then slice into 1/4-inch-thick slices (you should get 4–5 cups). Add fennel and garlic to pot and cook 2 minutes.
Add broth, bring to a boil, then reduce to a simmer over medium and cook until fennel is nearly tender, about 5 minutes.
Meanwhile make Parmesan Toasts. Heat broiler to high with rack set 6-inches from element. Arrange bread slices on a baking sheet and broil until lightly toasted, about 2 minutes. Flip bread and top with Parmesan, dividing cheese evenly between toasts. Broil toasts until Parmesan is melty and starting to brown, 2–3 minutes; set aside.
Stir kale, beans and milk into soup; cook until kale is wilted, about 2 minutes. Stir in vinegar and season with additional salt, red pepper flakes and vinegar to taste.
Serve soup with Parmesan toast and reserved fennel fronds