Heat the oil in a large skillet over medium-high. Add the sliced onion and saute, stirring occasionally, until onions begin to sweat and turn translucent, about 8 minutes. Sprinkle liberally with sea salt, reduce heat to medium-low, and continue cooking, stirring occasionally, until onions are golden-brown and caramelized, about 45 minutes more.
Preheat the oven to the high broil setting.
Slice the bagels in half and place them on a baking sheet. Toast on the top shelf of the oven under the broiler until golden-brown, about 1 minute. Remove bagels from the oven and add a slice of cheddar on all four slices of bagel. Place back in the hot oven until melted, about 1 to 2 minutes.
Smear 1 tablespoon of fig preserves over each cheesy bagel slice.
Top the bagel bottoms with wilted arugula, sliced apple, pear, caramelized onions, and a sunny side up egg. Place the bagel tops on top, slice in half and serve.
Notes
This sandwich is perfect for a hearty breakfast or brunch.