Melt 2 tablespoons butter in heavy medium skillet over medium heat. Add onion, brown sugar and vinegar; saute until onion is golden brown, stirring frequently, about 25 minutes. Remove from heat. Cool to room temperature.
Preheat oven to 350ºF. Sprinkle cheese over half of each tortilla, dividing equally. Sprinkle prosciutto and sauteed onion over cheese. Season with pepper. Fold other half of each tortilla over cheese mixture. Brush tortillas with some of melted butter. Brush heavy large skillet with some of melted butter. Place over medium-high heat. Working in batches, cook quesadillas just until brown spots appear, brushing skillet with butter between batches, about 2 minutes per side. Transfer quesadillas to heavy large baking sheet.
Bake until tortillas are golden and cheese melts, about 5 minutes.
Transfer quesadillas to work surface. Cut each into 6 triangles. Arrange on platter and serve hot.