Preheat the oven to 425ºF.
Add a swirl of oil, enough to coat the bottom of the pan, to a 10-inch skillet. Heat over medium heat until shimmering. Add the broccoli and shallots. Season with salt and pepper. Saute until the broccoli is tender, and the shallots are wilted and golden, 5 to 7 minutes, adding more oil as needed if the pan seems too dry. Set aside.
Roll out the pie crust, and fit it into a 9-inch pie plate (not deep dish). Crimp the crusts, or finish as desired.
Scatter the broccoli into the piecrust. Sprinkle most of the cheese top, reserving a few fingerfuls.
Combine the eggs and cream in a deep bowl. Whisk to blend. Season with salt. Pour into the pie dish. Tilt the pan to make sure the egg is evenly distributed. Sprinkle the remaining cheese on top.
Bake for 15 minutes. Reduce temperature to 325ºF. Bake until filling is set, and top is golden, about 20 to 25 minutes more. Let cool at least 10 minutes before slicing.