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For the popover: Preheat the oven to 425 degrees F. Place a 10-inch cast-iron skillet in the oven to heat. Prepare an ice bath in a large bowl and line a plate with paper towels.
Bring a pan of salted water to a boil, add the asparagus and cook until just tender, 1 to 2 minutes, depending on size. Drain, drop into the ice bath and dry on the prepared plate.
Using pot holders, carefully remove the pan from the oven. Add the butter to the hot pan and stir until melted. Arrange the asparagus in the bottom of the pan and then season with salt and pepper. Pour the egg mixture over the top. Bake until puffed and golden brown, about 15 minutes.
Remove the pan from the oven. Sprinkle the remaining fontina and the Parmesan on top, and put it back in the oven until the cheese is melted, 2 to 3 minutes. Remove from the oven and drizzle with some of the herb oil. Serve immediately.