Preheat the oven to 180 C / Gas 4. Line a baking tray with foil.
Arrange asparagus in an even layer on baking tray. Drizzle olive oil over spears and season with salt and pepper; toss and bake in preheated oven until asparagus is tender, about 10 minutes.
Meanwhile, cook diced bacon in a pan over medium-high heat until crispy, 4 to 5 minutes; set cooked bacon aside.
Slice the top of the asparagus spears off and set aside; puree asparagus stalks in a food processor until smooth.
Reduce oven temperature to 160 C / Gas 3. Lightly grease a baking dish with soft butter and sprinkle 1/2 tablespoon Parmesan cheese over the base.
Beat eggs with an electric mixer until foamy, 3 to 5 minutes. Carefully fold in milk, remaining Parmesan cheese, salt and pepper; stir in pureed asparagus stalks. Pour mixture into the prepared baking dish. Arrange asparagus tips on top in a decorative pattern.
Bake in the preheated oven until quiche is lightly browned on top and starts to pull away from the edges, 30 to 40 minutes. Cool dish for a few minutes and serve.