Preheat your grill.
Gently clean the mushrooms and remove the gills with a spoon. Brush all over with olive oil and set aside.
For the filling, mix the kale, cheese, onion, salt and pepper, and optional thyme together in a bowl.
Stuff the mushrooms, dividing the filling evenly among the mushrooms.
Turn the heat off on one side of the grill so you can cook the mushrooms over indirect heat. Place the mushrooms on the cool side of the grill, stuffed side up. Cover and cook until filling is bubbling and the mushrooms are well seared, 15-20 minutes. During the last couple minutes, top with additional cheese, if desired.
Remove from grill and serve immediately.