Add the potatoes to a pot of cold water and salt the water. Bring to a simmer over medium heat and cook the potatoes until tender, about 20 minutes. Drain the potatoes in a colander.
Add 2 tablespoons of olive oil to the pot over medium-high heat. Once the oil is hot, add the green onions and the kale. Saute until tender, tossing with tongs, about 3 to 4 minutes. Stir in the milk and butter and season with salt and pepper, to taste. Let the milk warm and the butter melt.
Put the potatoes into the pot and mash with a potato masher. Stir in the sour cream and the Parmesan and season with salt and pepper, to taste. Transfer to a serving bowl and serve.