Preheat oven to 425 degrees F (220 degrees C). Line a 9-inch quiche pan with parchment paper.
Press puff pastry into the quiche pan and trim to fit. Prick pastry with a fork and place in the refrigerator.
Bring a pot of water to a boil. Add Romanesco florets; cook until crisp-tender, about 3 minutes. Drain.
Melt butter in a large skillet over medium heat. Add mushrooms; cook and stir until lightly browned, about 5 minutes. Stir in drained Romanesco florets, oregano, salt, and pepper. Cook until florets are tender, about 5 minutes, adding water if they start to stick.
Beat eggs in a large bowl. Whisk in 1/2 cup plus 2 tablespoons milk and 1/2 cup plus 2 tablespoons heavy cream. Stir in mushroom mixture. Pour over pastry in the pan. Sprinkle 3/4 cup plus 1 tablespoon Swiss cheese on top.
Bake in the preheated oven until pastry is golden and quiche is set, about 30 minutes.