Heat oven to 350°F. In large bowl, whisk together eggs, milk, pecorino, 1/4 teaspoon salt and 1/2 teaspoon pepper. Heat oil in large oven-safe skillet (preferably cast-iron) on medium. Add onions and cook, covered, stirring occasionally, until tender, 5 minutes. Add sausage and cook, breaking up with spoon, until browned, about 5 minutes. Add kale and cook, stirring occasionally, until just wilted, about 1 minute. Reduce heat to low and add egg mixture, stirring to distribute sausage and vegetables. Transfer to oven and bake until almost set, 18 to 20 minutes. Remove from oven and heat broiler. Gently spread sauce over frittata, then top with mozzarella. Broil until cheese is browned and bubbling, 5 minutes. Top with basil and serve immediately.