Position a rack in the center of the oven and heat the oven to 375°F. In a food processor fitted with the slicing blade, slice the Brussels sprouts and shallots.
In a 12-inch oven-safe skillet, melt the butter over medium heat. Continue to cook the butter until it begins to brown and smell nutty. Set aside 1 Tbs. of the browned butter in a medium bowl.
Add the Brussels sprouts, shallots, 2 tsp. salt, and 1/2 tsp. pepper to the pan and toss to combine. Cook, stirring occasionally, until the Brussels sprouts and shallots begin to soften and brown in spots, about 6 minutes. Remove the pan from the heat.
Meanwhile, in a 2-quart saucepan over medium heat, combine the cream, Gruyère, nutmeg, cayenne, and 1/4 tsp. salt. Heat until the cheese is melted, whisking occasionally, about 4 minutes. Do not boil. Add the sauce to the Brussels sprouts, carefully stirring to combine.
Add the panko, Parmigiano, rosemary, and a pinch of salt to the reserved butter and mix thoroughly. Top the sprout mixture with the panko mixture.
Bake until the crumbs are browned and the Brussels sprouts are tender, 10 to 15 minutes. Let cool for about 5 minutes before serving.
For a more formal presentation, transfer the Brussels sprout mixture to a 12-inch-wide, 1-1/2-inch-deep ceramic baking dish before topping with the panko and baking.