These hors d'oeuvres are always a hit at parties, and they can be made ahead of time, which helps with planning. It's also possible to par-bake the mushrooms and freeze them; then you can simply transfer the frozen mushrooms to a baking sheet and bake until hot. Don't hesitate to experiment with the filling; there are endless delicious variations on the theme.
Heat the olive oil in a large skillet over medium heat. When hot, add the shallots and sauté for 3 minutes, stirring constantly. Add the garlic and prosciutto and cook another 2 minutes, taking care not to burn the garlic. Add the vermouth, lemon zest, and spinach, and turn the heat to medium-high. Cook until the spinach wilts, then add the crab, cream cheese, and half of the Pecorino. Season to taste with cayenne and salt, and let the mixture cool.
Remove the stems from the mushrooms and clean the caps with damp paper towels. Divide the filling between the mushrooms. Transfer the mushrooms to an oiled baking pan and sprinkle with the remaining cheese. Refrigerate until ready to cook.
Position a rack in the middle of the oven and preheat to 400 degrees F. Cook the mushrooms for 8 to 12 minutes, or until they're hot and the cheese has melted. Transfer to a serving platter with tongs. Serve warm.