Heat oil in a large nonstick skillet over med/high heat. Add the onions and season with salt. Cook for 15 minutes, stirring frequently, until onions are softened and beginning to brown.
Add brown sugar and vinegar and stir well. Reduce heat to medium and cook until onions are a deep brown, about 10 minutes. Stir often to prevent sticking and burning.
Make the Quesadillas:
Brush one side of each flour tortilla with olive oil. Place a tortilla oiled side down in a nonstick skillet & scatter with one third of each of the onions, cheese, prosciutto & arugula.