Make the caramelized onions:
Heat the vegetable oil in a medium saucepan over medium-low heat. Add the sliced onions, salt and pepper and cook, stirring occasionally, until the onions are very soft, about 10 minutes.
Add the sugar and continue cooking the onions for 5 more minutes.
Add the balsamic vinegar and reduce the heat to low. Continue cooking the onions, stirring occasionally, for about 15 minutes until they are very soft and dry.
Assemble the grilled cheese:
Slather one side of all four slices of bread with the butter and set them aside.
Assemble one of the grilled cheese sandwiches by stacking one slice of cheese atop one slice of bread (the non-buttered side) and topping it with half the caramelized onions, half the kale leaves and a second slice of cheese. Top it with another buttered slice of bread (buttered side on top). Repeat the assembly process for the second grilled cheese sandwich.
Heat a large skillet or sauté pan over medium heat. Place each sandwich on the skillet and cook until the bread is toasted and golden and the cheese begins to melt. Flip each sandwich once and continue cooking until the opposite side of bread is toasted. Slice and serve immediately.