Melt butter in a large skillet over medium-low heat. Mix onion with olive oil and add to skillet. Cook slowly until onions caramelize but do not brown, about 40 minutes. Stir occasionally. Season with salt and allow to cool, about 30 minutes.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler.
Lay out figs, cut side up, onto a baking sheet and sprinkle with sugar. Grill under the broiler until sugar has melted, 3 to 5 minutes. Allow to cool.
Combine chevre, herbed goat cheese, and milk in a small bowl until creamy.
Cut baguettes in half lengthwise and spread goat cheese mixture evenly over lower halves. Layer caramelized onions over the goat cheese, then cover with arugula. Cut figs into quarters or slices, depending on size, and layer onto the arugula. Top with the remaining baguette halves and wrap baguettes in plastic wrap. Refrigerate so flavors can combine, 2 to 3 hours. Slice and serve.