1. Cook bacon in a large skillet over medium heat, turning to brown both sides until crispy. Remove to a paper towel lined plate to drain. Chop or crumble bacon and set aside.
2. Remove all but 2 tablespoons of bacon grease from the pan and place over medium-low heat. Add the butter and onions and cook, stirring occasionally, until the onions are caramelized, 25 to 30 minutes. Season with 1/4 teaspoon salt and set aside.
3. Preheat the oven to 350°.
4. Carefully line a 9- inch pie plate with puff pastry. Carefully trim any overhang and fill any gaps with the trimmings. Just press the seams together. Crimp the edges with your fingers.
5. In a large bowl, whisk together the eggs, cream, Parmesan, 1 cup of the cheddar, dijon, remaining 1/2 teaspoon salt and pepper. Stir in the caramelized onions and bacon.
6. Pour into puff pastry and sprinkle with remaining 1/4 cup of cheddar.
7. Bake for 30-40 minutes, until the puff pastry is golden and the quiche is set. Check the quiche after 20 minutes, if the crust is browning too quickly, loosely cover it with foil.
8. Let the quiche rest for 10 minutes, then slice into wedges and serve with the sliced green onions on top.