Preheat the broiler and position the rack 8 inches from the heat. Crack the eggs into a large bowl. Add 2 tablespoons of grated Parmesan, season generously with salt and pepper and beat the eggs until blended.
In a large nonstick ovenproof skillet, heat 1 teaspoon of the olive oil. Add the sausage meat and onion and cook over moderately high heat, breaking the sausage up into small pieces, until browned and cooked through, about 5 minutes. Transfer to a plate. Add 2 tablespoons of the olive oil to the skillet and heat until shimmering. Add the potatoes and cook over moderately high heat, stirring occasionally, until crisp and cooked through, about 5 minutes. Add the kale, season with salt and cook, tossing, until softened, about 2 minutes longer.
Add the remaining 1 tablespoon of olive oil to the skillet and tilt the skillet to swirl the oil around the sides. Scatter the sausage and onion in the skillet. Stir the eggs and add them to the skillet. Cook over moderate heat for 1 minute. Gently lift the edge of the frittata and tilt the pan, allowing some of the uncooked egg to seep underneath. Cook until the bottom and sides are barely set, about 3 minutes.
Sprinkle the remaining 2 tablespoons of grated Parmesan on top and broil until the eggs are set and the top of the frittata is lightly browned, 1 to 2 minutes longer. Slide the frittata onto a large plate, cut into wedges and serve hot or warm.