Heat the oven to 375°F. Cook spaghetti in water with the tablespoon salt until 2 minutes short of the package time, then drain.
Meanwhile, season chicken with its salt and pepper. Heat oil in a large skillet and cook chicken for 7 to 8 minutes until at least 165°F; transfer to a plate. Cook mushrooms and onion in the skillet for 7 to 8 minutes until their water evaporates.
Add 2 tablespoons butter and flour and stir for 1 minute. Gradually stir in sherry, broth and milk and simmer for 4 minutes until lightly thickened, then fold in chicken, peas, Parmesan and spaghetti.
Transfer to a deep 9-by-13-inch, 4-quart baking dish. Melt remaining butter, toss with panko and scatter over the surface; cover the dish loosely with foil.
Bake for 20 minutes, then uncover for 10 to 12 minutes until the crumbs are golden and the center reaches 165°F. Rest for 5 minutes before serving.