Melt 2 tablespoons butter in heavy medium skillet over medium heat.
Add onion and sauté until onion is golden brown, stirring frequently, about 25 minutes.
Season with freshly cracked pepper.
Remove from heat. Cool to room temperature.
Brush heavy large skillet with remaining butter. Place over medium-high heat. Working in batches, Sprinkle cheese, prosciutto and sautéed onion on tortilla, dividing equally, top with another tortilla making a sandwich. Cook quesadillas just until brown spots appear, brushing skillet with butter between batches, about 2 minutes per side. Transfer quesadillas to cutting board. Cut each into 6 triangles. Arrange on platter and serve hot or at room temperature.
I usually serve with sour cream mixed with mango chutney and Tabasco, and lime wedges.