Place the skillet (stainless steel or cast-iron skillet work the best!) over medium/low heat and melt butter.
Add onion to the pan and stir to coat with butter.
Check and stir the onions every 5-10 minutes. If onions begin to burn, lower the heat.
Around 10 minutes onions will start to soften and turn translucent. Around 30 minutes the onions will begin to turn light brown in color. Around 50 minutes the onions will be dark golden and smell very caramelized. At this point, you can remove the onions from the heat.
Preheat the oven to 425F.
Scatter the onions evenly over each flatbread and sprinkle 1 tablespoon of feta cheese on each. Bake the flatbreads on a baking sheet for 8-10 minutes or until crisp. Remove and cool slightly.
Toss arugula with olive oil and salt and pepper to taste. Top each flatbread with equal amounts of arugula, slice of prosciutto and sliced figs. Sprinkle with additional feta, if desired.