Heat a medium skillet over medium heat and add the butter and olive oil.
Once the butter has melted, add the sliced onions and a pinch of salt. Cook, stirring occasionally, until the onions begin to soften, about 10 minutes.
Add the brown sugar and balsamic vinegar to the onions. Cook for another 10 to 15 minutes, stirring frequently, until the onions are deeply caramelized. Remove from heat and set aside. Take the toasted bagel halves and top each bottom half with a slice of spicy cheddar cheese, followed by a layer of pear slices, apple slices, and a generous spoonful of caramelized onions.
Top each half with another slice of cheddar cheese and cover with the top half of the bagel.
Place the prepared sandwiches on a skillet or griddle preheated to medium heat. Press gently with a spatula and fry for about 2 to 3 minutes per side, or until the cheese is melted and the bagel is golden brown.
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