Heat a large pan over medium heat. Add the olive oil and butter, letting them melt together.
Add the thinly sliced onions to the pan. Stir to coat the onions evenly in the oil and butter mixture.
Cook the onions on low heat, stirring occasionally, for about 10-12 minutes, until they are soft and golden.
Sprinkle in the brown sugar and stir well to dissolve.
Add the balsamic vinegar and cook for another 2-3 minutes, allowing the mixture to thicken slightly.
Season with salt and pepper to taste. Remove the onions from the heat and set aside.
Lay one slice of bread on a clean surface. Sprinkle half of the grated cheese evenly on top.
Spoon a generous layer of caramelised onions over the cheese.
Top with the remaining grated cheese and place the second slice of bread on top to form a sandwich.
Heat a pan over medium-low heat. Add a small amount of butter or olive oil to lightly coat the surface.
Place the assembled sandwich in the pan and press gently with a spatula.
Cook for 2-3 minutes on each side, or until the bread is golden brown and crispy, and the cheese is fully melted.
Slice the toastie in half, serve immediately, and enjoy the gooey, cheesy goodness!