Caramelize the Onions:

1. Heat 3 tablespoons of olive oil in a large skillet over medium heat.
2. Add the sliced onions and cook, stirring occasionally, for about 15-20 minutes, until the onions are golden brown and caramelized.
3. If the onions begin to stick, add a splash of water to deglaze the pan.

Cook the Eggs:

1. In the same skillet, cook the eggs sunny-side-up to your desired doneness.
2. Remove from heat and set aside.

Toast the Bagels:

1. While the onions and eggs are cooking, toast the everything bagels until golden and crispy. Set aside.

Assemble the Sandwiches:

1. Spread 2 tablespoons of fig preserves on the bottom half of each toasted bagel.
2. Layer each bagel with a slice of Cheddar cheese, followed by the caramelized onions, arugula, apple slices, and pear slices.
3. Place the cooked egg on top and finish with another slice of Cheddar cheese.
4. Place the top half of the bagel on the sandwich to close.

Serve:

1. Serve the sandwiches warm and enjoy the wonderful contrast of flavors from the sweet fruit, tangy fig preserves, savory caramelized onions, and melty cheddar.