Preheat the oven - Preheat the oven to 375°F (190°C).

Prepare the pie crust - Roll out the pie crust on a floured surface to fit a 9-inch deep dish pie dish. Carefully transfer the crust to the pie dish and trim any excess dough. Use a fork to prick the bottom of the crust several times to prevent it from puffing up during baking.

Blanch the asparagus - Bring a large pot of salted water to a boil. Add the asparagus and blanch for 2-3 minutes, until bright green and slightly tender. Drain and rinse the asparagus under cold water to stop the cooking process. Pat dry with a paper towel.

Saute the shallot - Melt 1 tablespoon of butter in a frying pan over medium heat. Add the chopped shallot and sauté for 2-3 minutes, until softened and lightly browned. Remove from heat and set aside.

Whisk the filling - In a large mixing bowl, whisk together the eggs, heavy cream, whole milk, salt, and black pepper until smooth and well combined.

Assemble the quiche - Spread the sautéed shallot evenly over the bottom of the prepared pie crust. Arrange the blanched asparagus spears on top of the shallots. Sprinkle the shredded Gruyere cheese over the asparagus. Carefully pour the egg mixture over the filling, making sure that it's evenly distributed.

Bake the quiche - Place the quiche on a baking sheet and bake in the preheated oven for 40-45 minutes, until the filling is set and the crust is golden brown. Remove from the oven and let cool for 10 minutes before slicing and serving.