In a cast-iron skillet over medium heat, or on the grill, cook burger patties until you see juices form on top, about 5 to 6 minutes or so, then flip and top with brie to melt. Continue cooking another 5 minutes or so, or until patty reaches desired doneness.
In a large cast-iron skillet over medium-low heat, caramelize the onions in the ghee. Sauté the onions until they start to brown, then periodically add just enough water to cover the bottom of the skillet and stir every so often, adding more water as needed. Repeat until onions are browned and caramelized to your liking. This will take several minutes. Season with sea salt, to taste if desired.
While the onions cook, in a separate small skillet or saucepan, cook the blueberries in just enough water to cover, smooshing the berries with a spatula as they warm and soften, eventually creating a compote as the water evaporates. Remove from heat. If desired, you can stir in a drizzle of honey or maple syrup to sweeten further.
To serve, top bun with the cooked patty, the arugula, the caramelized onion and the blueberry compote. Serve warm.