Preheat the oven to 400° F and place an oven rack in the center of the oven.
Heat a 10-inch skillet (cast iron or non stick works best) over medium heat. Add the oil and twirl the pan to evenly coat, making sure to coat the sides of the skillet. Add the onions, and cook for about 3 minutes, or until they wilt and are transparent. Stir in the sausage and cook for another two minutes, followed by the kale.
Quickly whisk together the eggs, milk, salt, and pepper, and pour the eggs into the skillet. Using a spatula, move the edges of the eggs towards the center of the pan, stirring around the curds as they form.
Gently stir in the cheese.
Place in the oven to bake for about 10-15 minutes, or until the frittata is lofty and the center is no longer runny when you insert a knife.
Remove from the oven, let it stand for a few minutes before slicing and serving.