How to Make It
Thinly slice 1 lemon half and cut other half into wedges; set aside. Finely chop anchovies, garlic, and oregano together on a cutting board. Drag the flat side of knife back and forth along mixture until a paste forms.
Heat 1 tablespoon oil in a large, high-sided skillet over medium-high. Add anchovy paste; cook, stirring, until fragrant, about 1 minute. Stir in potatoes and broth and bring to a boil over high. Reduce heat to medium; simmer, covered, until potatoes are tender when pierced with a fork, about 12 minutes.
Meanwhile, season fish with ½ teaspoon each salt and pepper. Heat remaining 1 tablespoon oil in a large nonstick skillet over medium-high. Add fish; cook on 1 side until browned, about 4 minutes. Transfer to a plate.
Uncover potato mixture and stir in kale. Arrange fish fillets, browned sides up, on top. Return to a simmer and cook, uncovered, until fish flakes with a fork and kale is wilted, about 5 minutes. Using a slotted spoon, transfer fish, potatoes, and kale to 4 shallow bowls.
Increase heat to high and add lemon slices to skillet. Boil until broth is reduced by half, about 4 minutes. Stir in remaining ¼ teaspoon salt and ½ teaspoon pepper. Ladle broth over fish and vegetables; garnish with chives and lemon wedges.