Position a rack in the center of the oven and heat the oven to 425°F. Trim and discard the very bottom of the mushroom stems. Remove the mushroom stems and finely chop them, along with five of the largest mushroom caps.
Heat a medium sauté pan over medium heat for 1 minute and add 2 Tbs. of the butter. When it has melted, add the pancetta and cook until it starts to render some of its fat, 1 to 2 minutes. Add the shallots, sage, and chile flakes; cook gently until the shallots are tender, about 4 minutes (reduce the heat if the shallots begin to brown). Stir in the chopped mushroom stems and 1/2 tsp. salt. Cook, stirring frequently, until the mixture is tender, about 3 minutes. Add the remaining 1 Tbs. butter. When it has melted, transfer the mushroom mixture to a bowl and stir in the breadcrumbs and Parmigiano. Season to taste with salt and pepper and let cool slightly.
Butter a shallow baking dish large enough to hold the mushrooms in one layer. Arrange the mushrooms in the dish and season the cavities with salt. Stuff each cavity with a rounded teaspoonful of the filling, or more as needed. The filling should form a tall mound. (You may have leftover filling; if you have extra mushrooms, keep stuffing until you run out of filling.) Drizzle the mushrooms with the olive oil and bake until the mushrooms are tender and the breadcrumbs are golden brown, 20 to 25 minutes. Transfer to a platter and serve warm.
For a crisp, golden-brown stuffing, drizzle with olive oil just before baking.