Preheat your oven to 350º F. Chop the kale and place it in a pan with the olive oil and vegetable broth and cook over medium-high heat until all of the liquid is absorbed (about 8 minutes). Once cooked, set aside.
While the kale is cooking, prep the pattypan squash by cutting the stems off the top and the bottom, removing as little of the flesh of the squash as possible. Then scoop out the seedy center of each squash, leaving a "cup" to place your stuffing into.
To prepare your stuffing, mix together your kale, corn, white onion, garlic, salt, pepper and 1 C of cotija cheese in a bowl.
Lightly grease a large casserole dish with olive oil or cooking spray and evenly place your pattypan in the dish, cup side up. To stuff the pattypan squash, squeeze together a ball of stuffing in your hands (as if you are making a snowball), and place it into one pattypan cup. The balls will be approximately the size of a medium ice cream scoop. Continue doing this until all of the pattypans are filled.
Bake at 350º F for 35-45 minutes. Cooking for a shorter amount of time will yield a firmer squash and cooking for 45 minutes with result in a softer squash.
Once cooked, remove the pattypans from the oven and sprinkle them with cotija cheese and chiffonade basil.