Trim woody ends from asparagus spears; cut in half if necessary to fit your pot.
Bring a pot of generously salted water to a boil over high heat. Add asparagus and boil until slightly tender, 2 to 3 minutes. Use a slotted spoon to transfer asparagus to a bowl of very cold water; let soak until completely cool.
Drain asparagus well, then cut into very small pieces. Transfer to a bowl and season with salt and pepper. Add bread crumbs, Pecorino Romano, and cayenne pepper. Mix in eggs, one at a time, until well combined.
Heat olive oil in a skillet over medium heat. Working in several batches, scoop some asparagus mixture into the hot skillet. Flatten mixture with a fork and cook until the bottom is browned, about 4 minutes. Flip and brown the other side, about 4 minutes more. Repeat to cook remaining patties.