Place large skillet over medium heat and add olive oil and 2 Tbsp butter.
Once butter is melted, add in onion slices, sprinkle with salt and pepper, and cook for about 5 minutes, stirring frequently.
After 5 minutes, turn heat to low, add brown sugar and apple cider vinegar and stir occasionally as onions begin to brown.
Continue to cook, stirring occasionally until onions are browned to your liking. Caramelization process will likely take at least 30 minutes total. Before removing skillet from heat, stir in thyme.
Once onions are caramelized, spread remaining Tbsp butter on one side of each bread slice.
Place a few slices of cheese on two of the bread slices (the unbuttered sides). Pile caramelized onions on top of cheese and add remaining cheese slices to the top of the onions. Place remaining slices of bread on top, buttered sides facing out.
Heat a large skillet or griddle over medium-low heat (if you want to use the same skillet you just cooked the onions in, you can) and place sandwiches on skillet.
Let cook for about 2 minutes before flipping sandwich and cooking for another 2 minutes on the other side, until bread is golden and cheese is melty.
Remove from skillet and slice sandwiches in half before serving.