Preheat oven to 500 degrees. Trim bottom of broccoli. Cut stems into slices and microwave in covered dish until soft but still bright green, about 4 minutes. Chop in food processor. Repeat with florets, but microwave about 3 minutes.
Whisk eggs in large bowl; add milk. In another large bowl, toss cheeses, flour, herbs, and spices. Pour egg-milk mixture over cheeses and stir. Add broccoli. Pour filling into unbaked pie shell.
Place on bottom rack of oven and bake 10 minutes. Reduce oven temperature to 350 degrees and bake 40 minutes longer. Let stand 10 minutes before cutting.