Season fillets on both sides with salt and pepper. Heat 1 tablespoon of oil in a medium nonstick, lidded sauté pan or skillet over medium flame. Place the fillets in the pan flesh side (pretty side, some would call it) down and cook until lightly browned, 3 to 4 minutes.
Turn and cook on other side for about 2 minutes. Transfer to a plate. The fish won't be cooked through at this point.
Purée 1 of the cans of beans with some of the wine and water in a blender.
Wipe pan clean with paper towel and add another tablespoon of oil. Cook shallot, stirring frequently to avoid overly browning, until softened, about 3 minutes. Add beans and puréed beans, wine and water to pan, stirring to combine.
Add kale to pan in handfuls and cover. Cook for about 2 minutes, allowing the alarming volume of kale to wilt. Stir to work kale into bean mixture; it will continue to wilt and collapse.
Season everything lightly with salt and pepper. Lay tilapia fillets on top of bean mixture and sprinkle capers over everything. Cover pan and cook until tilapia is just cooked through, 3 to 4 minutes. (The capers' fragrance will suddenly get frighteningly big—don't worry, it will subside.)
Transfer tilapia fillets to a small plate. Taste beans and kale and adjust seasonings. Add a little water if the beans have gotten too thick. Divide beans and kale between two shallow bowls and top each with a tilapia fillet. Serve.