On a work surface, layer salami on half the tortilla. In a large nonstick skillet, heat 1 teaspoon oil over medium-high. Add mushrooms and cook, stirring occasionally, until golden brown, 8 minutes. Season with salt and pepper.

Place mushrooms and cheese on top of salami; fold tortilla over filling. Place quesadilla in skillet and cook over medium until tortilla is golden brown and the cheese is melted, about 3 minutes per side.Transfer quesadilla to a cutting board and cut into 3 wedges.

In a small bowl, toss arugula with lemon juice and 1/4 teaspoon of oil. Season with salt and pepper and serve alongside quesadilla.