Line a small plate with a paper towel. Heat 1 tablespoon of the oil in a small skillet on medium-high heat. Add capers and fry for just a few seconds, until the buds open and they become crispy. Quickly remove with a slotted spoon and drain on paper towel.

Meanwhile, bring a large pot of salted water to a boil over high heat. Have a large container of ice water ready. Add the asparagus to the boiling water and let boil for 3 minutes until tender-crisp but not limp. Remove the asparagus with tongs (keep water hot for reheating) and transfer to the ice bath to stop cooking. Drain on paper towels.

Place a cast-iron skillet large enough to accommodate both fillets on medium heat. Coat the fillets liberally with the vegetable oil spray. Spread the grated cheese on a plate and dredge the skinless (or rounded) side of the fillets in it, coating well.

Add the remaining 2 tablespoons of oil to the hot pan then place the fish, cheese side down, in the hot oil. Do not disturb for about 4 minutes.

Put about a tablespoon of the cheese directly into the skillet as a test at the same time you add the fish. When you can slide a spatula under that fried cheese medallion, you can carefully loosen the fillets and turn them. A golden brown, fried cheese crust that completely coats the fish is what you’re looking for here. (Lower the heat if it’s browning too quickly.) Cook about 2 minutes on the uncoated side.

While the fillets cook, return the asparagus to the pot of hot water for another minute or so to reheat. Move the fillets to a warm plate. Wipe the pan out with a paper towel; immediately return it to the heat and add the butter. When the butter is medium brown (take care not to burn), add the lemon juice, parsley, and crabmeat. Turn off heat.

Using tongs, divide the hot, drained asparagus between 2 heated plates. Place the fish on top of the asparagus spears, spoon the brown butter and crabmeat mixture over the fish, and sprinkle with the fried capers.