Melt the butter in a skillet over medium-low heat.
Add the sliced onions and ¼ teaspoon salt and cook, stirring occasionally, for 40 minutes, or until the onions are deeply caramel in color and soft.
In the meantime, heat 2 tablespoons of oil in a skillet over medium heat.
Add the mushrooms and cook until glossy and browned, about 5 minutes. Sprinkle with ¼ teaspoon of salt and 1 teaspoon of pepper, then remove and reserve.
Add the ground beef to the pan and season it with ½ teaspoon salt and pepper. Cook, crumbling until browned, about 8 minutes.
In a small bowl, stir together the mayonnaise, garlic, ¼ teaspoon salt, ¼ teaspoon pepper, and lemon juice.
To assemble the bowls, first divide the arugula and watercress between 4 bowls.
Add the ground beef, mushrooms, and caramelized onions to the bowls, then divide the Swiss cheese between the bowls.