Heat the oil in a medium saucepan over medium heat until shimmering. Add the garlic and cook until fragrant, about 1 minute. Stir in the kale and 1/4 cup water. When the water boils, reduce the heat to low, cover and cook until the kale is tender, about 15 minutes.
In a large bowl, mix together the crabmeat, 1/2 cup of the Cheddar, 1/4 cup of the Parmesan and the sour cream, cream cheese, mayonnaise, salt and pepper. Stir in the cooked kale.
Transfer the mixture to an 8-by-8-inch baking dish. Sprinkle with the remaining 1/4 cup Cheddar and 2 tablespoons Parmesan. Bake until the dip is hot and bubbly and the cheese on top is melted and golden, 35 minutes.
Lacinato kale is also known as Tuscan or dinosaur kale.