Pat scallops dry and toss with about half the salt. Set scallops in the refrigerator until you're ready to use them.
In a large stainless steel pan, heat 1 tablespoon butter over medium heat. When the butter is melted, but before it begins to brown, add the sliced shallot. Sauté 2-3 minutes, or until the shallots begin to brown.
Push shallots to the edge of the pan (or remove shallots and reserve them), and add sliced mushrooms. Sauté mushrooms until they're browned, about 4 minutes.
Remove mushrooms and shallots, and divide between two ramekins. Set aside.
Add remaining 1 tablespoon butter to the pan, and lower heat to medium low. Whisk in milk, and then whisk in the pumpkin. Continue to whisk or stir until the sauce is smooth and creamy (about 1 minute).
Season the sauce to taste with remaining salt, cayenne, and nutmeg. (Make sure to actually taste the sauce-- if it tastes bland, add a little more salt. If you want it a little spicier, add more cayenne or nutmeg.) Remove the sauce from the heat.
Move an oven rack to the top of your oven, and turn on the broiler.
Set scallops in the ramekins, on top of the mushroom and shallot mixture. Pour the pumpkin sauce into the ramekins, covering the SIDES of the scallops, but not the tops.
Divide the Parmesan on top of both ramekins. Set ramekins on a baking sheet, and then slide it onto the top oven rack under the hot broiler.
Broil the scallops for 5-7 minutes until the cheese has turned golden and is bubbling. (Watch the scallops carefully after about 5 minutes-- you want them golden, not burnt.)
Allow the ramekins to cool slightly, top with fresh chives (if desired) and serve your broiled scallop gratin while it's still hot.
Make Ahead:This dish can be completely assembled up to 1 day ahead, but wait to broil it until just before serving.