Place 1 tortilla on a work surface. Layer a fourth of the pear slices, a fourth of the prosciutto, ½ cup arugula, and a fourth of the cheese over half the tortilla. Fold tortilla over filling. Repeat to make remaining quesadillas.

Heat 1 tablespoon oil in a large nonstick skillet over medium-high. Add 2 quesadillas to skillet; cook, flipping once, until tortillas are browned on both sides and cheese is melted, about 2 minutes per side. Remove from skillet. Repeat with 1 tablespoon oil and remaining quesadillas.

Whisk lemon juice, salt, pepper, and remaining 1 ½ tablespoons oil in a large bowl. Add greens and toss to coat. Serve immediately alongside quesadillas.