Heat 2 tablespoons oil in a heavy large skillet over medium-low heat. Add the onions and sugar, and cook until the onions are golden brown, about 20 minutes. Season the onions, to taste, with salt and pepper; set aside.

Preheat the oven to 250 degrees F.

Sprinkle some Fontina over half of a tortilla. Arrange some bell peppers and sauteed onions on top of the Fontina. Sprinkle with chopped parsley and Parmesan. Sprinkle more Fontina over. Fold the clean half over the filling to cover and press gently to seal. Repeat with remaining tortillas evenly dividing the filling ingredients.

Heat 1 tablespoon oil on a heavy large griddle over medium heat. Working in batches, cook the quesadillas until the cheese melts and the tortillas are golden brown, adding more oil as needed, about 3 minutes per side. Transfer the quesadillas to baking sheets and keep them warm in the oven while cooking the remaining quesadillas.

Sprinkle the remaining Parmesan over the quesadillas. Cut each quesadilla into 4 wedges. Arrange them on a platter. Garnish the platter with parsley sprigs and serve.

Do-Ahead Tip: The quesadillas can be assembled up to 8 hours ahead. Arrange them on a baking sheet, cover tightly with plastic wrap, and refrigerate.