Preheat oven to 350 degrees 15 minutes before baking.
Melt 2 tablespoons butter in a heavy medium skillet over medium heat.
Add onion, brown sugar and vinegar, sauté until onion is golden brown, stirring occasionally, about 25 minutes, lowering heat if necessary to prevent over-browning.
Remove from heat.
Cool to room temperature.
Sprinkle cheese over half of each tortilla, dividing equally.
Sprinkle prosciutto and sautéed onion over cheese.
Season with pepper.
Fold other half of each tortilla over cheese mixture.
Melt remaining 2 tablespoons of butter.
Brush tortilla with some of the melted butter.
Brush a heavy, large skillet with some of the melted butter.
Place over medium-high heat.
Working in batches, cook quesadillas just until brown spots appear, about 2 minutes per side.
Brush skillet with butter between batches.
Transfer quesadillas to heavy, large baking sheet.
Bake until tortillas are golden and cheese melts, about 5 minutes.
Transfer quesadillas to a work surface.
Cut each into 6 triangles.
Arrange on a platter and serve hot.