Preheat oven to 450 degrees F. Fit piecrust into deep-dish pie plate, flute as desired. Prepare for blind baking using either method:
Preheat oil in a pan, add onion and broccoli and saute for two or three minutes. Add about 3 tablespoons water, add lid and cook until broccoli is done, just a bit past the tender/crisp stage, stirring once or twice. If there is any remaining water, drain.
Add cheese first, then the ham to the pie shell, top with the broccoli/onion mixture (just pile the broccoli up and adjust it after adding the filling) and fill with the quiche mixture, below.
In a medium-sized bowl, add eggs, whisk, then add the milk. Whisk together and add salt and pepper.
Pour liquid into the crust with the cheese, ham & broccoli mixture. Place in oven. Turn heat down to 350 degrees and continue to bake until the quiche is golden and slightly browned in spots and when shaken, jiggles just a little bit in the very center of the quiche, 50 minutes to an hour or so.
Remove quiche from oven. Quiche will continue to bake just a bit as it sits and will firm up as it cools. Cool, preferably on a rack on the counter. If refrigerating warm quiche, cover loosely until cold (to prevent condensation) and then cover tightly.
Serve warm, room temperature or cold.
Note: leftover broccoli works well in this recipe and saves a step.