In a 10- to 12-inch frying pan over high heat, stir mushrooms and onion in 2 tablespoons oil until they begin to brown, 7 to 8 minutes.
Add salt, nutmeg, allspice, cinnamon, and cayenne; stir until fragrant, about 2 minutes. Add vinegar and molasses; stir often until liquid is evaporated. Scrape into a large bowl. Add flour and mix to blend, then stir often until lukewarm, about 5 minutes.
Add turkey, broth, and egg to bowl; mix well. Gently stir in pickled onions.
Oil a 5- by 9-inch nonstick loaf pan (2 qt.). Line pan neatly with a single layer of prosciutto slices, overlapping edges slightly; press ends of prosciutto against pan sides up to, but not over, rim. Scrape meat mixture into pan and gently pat or spread level. Fold ends of prosciutto slices neatly over meat.
Bake turkey loaf in a 350° regular or convection oven until a thermometer inserted in center of thickest part reaches 160° and meat is no longer pink (cut to test), about 55 minutes; loaf will begin to shrink from pan sides.
Let stand at room temperature for at least 10 minutes. Invert a slightly larger, rimmed platter over pan; hold pan and platter together and invert. Lift pan off to release loaf.
Nutritional analysis per serving made with turkey breast.