Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with aluminum foil.
Place pie crust into a 9-inch deep pie dish and prick crust all over with a fork; brush with oil. Place crust onto prepared baking sheet.
Bake crust in the preheated oven for 8 minutes.
Place broccoli into a bowl and microwave on high until broccoli is bright green and slightly tender, about 4 minutes. Whisk eggs, milk, Dijon mustard, salt, pepper, and thyme in a bowl; stir broccoli into egg mixture. Pour mixture into crust and sprinkle quiche with Cheddar and Parmesan cheeses. Place pie dish on the baking sheet.
Bake until a knife inserted into the middle of the quiche comes out clean, about 40 minutes. Let stand 10 minutes before cutting.