In a non-stick skillet, caramelize the onions in the butter, about 20 minutes. Season with salt and pepper. Set aside.
With the rack in the middle position, preheat the oven to 180 °C (350 °F).
In a pot of boiling water, cook the sausage for about 5 minutes. Drain. Warm the bread in the oven.
Preheat the oven’s broiler.
Open the buns horizontally and spread mustard inside. Cover with caramelized onions and add the sausages. Sprinkle with cheese. Place the hot dogs on a baking sheet. Melt the cheese in the oven. Garnish with arugula.