Heat the oven. Place a rack in the middle and heat the oven and a large sheet pan to 220°C (425°F).
Prep and roast the parsnips (about 20 minutes). Peel the parsnips and cut them into sticks. If the thick tops have a woody core, cut it out. Toss with the oil, salt, and pepper, then spread them on the hot pan in a single layer. Roast, tossing once, until the edges are golden brown and a knife slips in easily. Published recipes give a wide range of 20 to 40 minutes depending on size, so begin checking at 20.
Mix the crab cakes (about 5 minutes). Whisk the egg, mayonnaise, mustard, Worcestershire, and seasoning in a large bowl. Stir in the green onions, celery, parsley, and panko. Gently fold in the crab, keeping the lumps intact. The mixture should hold together when squeezed, and if it’s loose, add 1 to 2 tbsp more panko.
Shape and wrap (about 5 minutes). Divide the mixture into 4 cakes about 3 inches (7.5 cm) wide and 1 inch (2.5 cm) thick. Wrap 2 slices of prosciutto around the side of each cake, overlapping the ends and pressing gently so it clings.
Chill (10 minutes). Place the cakes on a plate in the refrigerator while the parsnips finish. Published recipes chill for 30 minutes, so this short chill is my compromise. If you have the time, chill longer.
Mix the sauce. Stir the mayonnaise, mustard, and lemon juice together.
Sear (6 minutes). Heat the oil in an ovenproof skillet over medium-high heat until it shimmers but doesn’t smoke. Add the cakes without letting them touch. Cook 3 minutes, until the bottom is golden and the prosciutto starts to crisp, then flip carefully and cook 3 minutes more.
Finish in the oven (4 to 5 minutes). Take the parsnips out, then slide the skillet into the oven. The cakes are done when the center reads 71°C (160°F), the federal figure for egg dishes, which I use because the binder contains raw egg. The skillet handle will be very hot.
Serve. Plate each cake with a pile of parsnips, a spoonful of sauce, and a lemon wedge.