Melt 2 Tablespoons butter in heavy medium skillet. Add onion, brown sugar and vinegar; sauté over medium heat until onion is golden, stirring frequently, about 30 minutes. Cool to room temperature.
Preheat grill. Sprinkle cheese over half of each tortilla. Sprinkle prosciutto and caramelized onions over cheese. Season with pepper. Fold other half of each tortilla over cheese mixture. Brush tortillas with melted butter.
Grill quesadillas until grill marks begin to appear and cheese is melted, Transfer quesadillas to work surface. Cut each into 4 triangles. Arrange on platter and serve hot.