Preheat the oven to 200 degrees F. Turn the griddle to medium-low and heat the oil. Add the onions and sprinkle with some salt and pepper. Add the bratwurst around the onions and cook until the onions are slightly charred on the bottom, 3 to 4 minutes. Turn the onions (don’t worry if the rings come apart), sprinkle with some salt and pepper and cook until the second side is charred and the onions are crisp-tender, 4 minutes. Transfer the onions to a medium bowl and set aside.
Keep an eye on the brats while the onions are cooking; turn them frequently until they are deep brown all over and an instant-read thermometer inserted in the center registers 165 degrees F, about 20 minutes total. Transfer the brats to a small rimmed baking sheet and keep warm in the oven. While they are browning, butter the insides of the buns.
Put the kale on the griddle and turn once or twice with tongs to coat with the brat drippings. Season lightly with salt and keep turning until the kale is tender, about 2 minutes. Transfer to the bowl with the onions.
Wipe the griddle clean with a paper towel. Add the buns butter-side down and cook until golden and toasted, about 2 minutes. To serve, spread mustard on the buns, add the brats and top with the kale and onions.