1. Preheat oven to 350 degrees. Lightly oil a rimmed baking sheet. Arrange portobello caps, gill sides down, on sheet. Bake until tender, 20 to 25 minutes. Transfer to a plate to cool. Preheat broiler.

2. Stir together cheese, breadcrumbs, 1 tablespoon parsley, 1 tablespoon chives, and 1 tablespoon oil; set aside.

3. Chop portobello stems into 1/2-inch pieces. Heat remaining tablespoon oil in a large skillet over medium heat until hot but not smoking. Add shallots; cook, stirring, until softened, about 2 minutes. Add sliced mushrooms and chopped stems; cook, stirring occasionally, until tender, 6 to 7 minutes. Add wine; cook until most of the liquid has evaporated, about 2 minutes. Stir in cream, remaining 2 tablespoons each parsley and chives, and the salt and pepper. Remove from heat.

4. Arrange portobello caps, gill sides up, on a clean baking sheet. Divide mushroom mixture and then crumb mixture among caps. Broil until bubbling and golden brown, about 2 minutes.