Not too messy. More of a handheld snack.
Heat olive oil over medium heat, in a large non-stick pan of at least 10 inches in diameter.
Add sliced mushrooms and cook until soft and lightly caramelized, about 4 to 5 minutes. Lower heat and add cannellini beans. Continue cooking for another 5 minutes until beans are softened. Season with salt and pepper. Remove mixture to a medium-sized bowl and gently mash beans with a fork.
If necessary, wipe out pan with a dampened towel. Lightly oil pan and reheat over medium heat. Place one tortilla in pan and sprinkle half the cheese over the entire surface. Spread half the arugula and bean mixture over just one-half of tortilla.
Fold the tortilla in half and press down lightly. Cook for about 1 to 2 minutes per side, until golden brown. Transfer quesadilla to a cutting board and repeat process with second tortilla.
Cut each quesadilla into four equal wedges. Serve warm with suggested dips.