Preheat the oven to 350°F. Fit the pie crust into a 9 to 10-inch deep-dish pie pan and crimp the edges. Refrigerate while you prepare the filling.
Cook the bacon in a large skillet over medium heat until crispy. Remove with a slotted spoon to a paper towel-lined plate. Reserve about 2 tablespoons of bacon drippings in the skillet.
Add the diced onion to the skillet with the bacon drippings. Cook over medium-low heat for 20 to 25 minutes, stirring occasionally, until deeply caramelized. Remove from heat and let cool slightly.
In a large bowl, whisk together the eggs, heavy cream, salt, and pepper until fully combined.
Stir in 1¾ cups of the shredded cheddar cheese, the cooked bacon, and the caramelized onions. Stir until evenly distributed.
Remove the pie crust from the refrigerator. Pour the filling into the crust. Scatter the remaining ¼ cup of cheese over the top.
Place the pie dish on a rimmed baking sheet. Bake for 50 to 55 minutes, until the custard is set and the top is golden. The center should have a slight jiggle.
Let the quiche rest for at least 15 minutes before slicing. Garnish with sliced green onions and serve warm or at room temperature.