Heat oven to 375 degrees F. Fit crust into a 9-inch pie plate, cover loosely with plastic wrap and refrigerate.
Heat oil in a large sauté pan over medium heat. Add onions and cook until browned and soft, stirring once in a while, about 35 to 40 minutes. (If onions begin to burn, reduce heat to medium-low.) Set aside to cool a bit.
In a bowl, whisk together eggs, milk, mustard, salt and nutmeg. Remove pie plate from refrigerator. Sprinkle cheese evenly on bottom of crust. Scatter onions on top. Pour egg mixture over cheese and onions. Bake quiche at 375 degrees F for 45 minutes, or until eggs are set and crust is browned. Cool 10 minutes on a wire rack. Serve with arugula, if desired.