In a large skillet, heat 1 tablespoon of the oil and the butter over medium heat.
Add the onions and sugar.
Cook for 6 to 7 minutes, stirring occasionally until lightly browned.
Add the garlic and cook for 2 minutes longer.
Transfer to a bowl and set aside.
Return the skillet to the stove and heat the remaining 1 tablespoon of oil.
Add the mushrooms, thyme, salt, and pepper.
Cook until the mushrooms are browned and tender.
That’s about 7 to 8 minutes.
Stir in the onions and cook for 2 extra minutes.
Remove from the heat, allow to cool for 3 minutes, and stir in the arugula.