1. Place morels in a bowl and add 2 cups boiling water. Let stand until soft, about 15 minutes. Gently squeeze mushrooms in water to release grit. Lift from water and squeeze out most of the liquid. Set mushrooms aside. Without disturbing grit, carefully pour reserved soaking liquid into another container. Discard grit.
2. Meanwhile, cut 4 slices of prosciutto into thin slivers. In a 5- to 6-quart pan, stir slivers over high heat until crisp and lightly browned, 5 to 8 minutes. Remove from pan; when cool, wrap airtight and chill up to 1 day.
3. Add butter and shallots to pan. Stir over high heat until shallots are limp, about 2 minutes. Add flour and stir until light golden brown, about 2 minutes. Remove from heat and whisk in mushroom liquid, sherry, broth, and cream, then add morels. Stir often over high heat until sauce is reduced to 2 1/2 cups, 20 to 25 minutes.
4. Cut polenta into chunks, then mash with hands or a potato masher until it's fine crumbs. Divide polenta evenly between 2 casseroles, each 3 quarts (about 9 by 13 in. and at least 2 in. deep). Press polenta into an even layer.
5. Lay a chicken breast half and 2 asparagus spears on a prosciutto slice and wrap to enclose. Place chicken, asparagus side up, on polenta. Repeat to wrap remaining chicken and asparagus, and space pieces evenly apart in casseroles. With a slotted spoon, scatter morels around chicken. Pour sauce equally over chicken in each casserole. If making ahead, cover and chill up to 1 day.
6. Bake, covered, in a 400° oven until chicken breasts are no longer pink in center of thickest part (cut to test), 30 to 35 minutes (35 to 40 minutes if chilled). Uncover and sprinkle with cheese, then prosciutto shreds. Bake until cheese melts, 2 to 3 minutes more. Serve with salt to add to taste.