Heat 2 Tbsp of oilve oil in a large, heavy skillet over medium-low heat. Add thinly sliced onions and sugar; cook until the onions are golden brown (about 20 minutes). Season the onions with salt and pepper; set aside.
Preheat the oven to 250 degrees.
Sprinkle some Fontina cheese over half of a tortilla. Arrange some bell peppers and sauteed onions on top of the Fontina. Sprinkle with chopped parsley and Parmesan. Sprinkle with more Fontina. 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 Tbsp of oil on a large heavy 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 three minutes per side. Transfer the quesadillas to baking sheets and keep them warm in the oven while cooking the remaining quesadillas.
Sprinle the remaining Parmesan over the quesadillas. Cut each quesadillo into four wedges. Arrange them on a platter. Garnish with parsley sprigs and serve.
Do-ahead step: The quesadillas can be assembled eight (8) hours in advance. Arrange them on a baking sheet, cover tightly with plastic wrap and refrigerate.
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