Preheat broiler, setting temperature to high. Set oven rack in the middle of the oven. Line a baking sheet with foil.
Brush the mushroom caps with olive oil on both sides. Sprinkle with ½ teaspoon kosher salt and ⅛ teaspoon black pepper. Broil 5 minutes on each side, or until just tender.
Remove mushrooms from oven. Heat 1 tablespoon olive oil in a large skillet over medium heat, about 3 minutes. Add the onion and cook 5-7 minutes, stirring occasionally, until golden. Add the garlic, spigarello, ½ teaspoon kosher salt and ⅛ teaspoon black pepper and cook, stirring to combine everything, 1-2 more minutes. Remove from heat and allow to cool a few minutes, then mix in the Parmesan.
Evenly distribute the spigarello mixture among the mushrooms. Don't press down on the filling - it's prettier when it's piled high on top of the mushrooms. Place back under the broiler for 2-3 minutes, or until filling is golden.
Stuffed and baked portobello mushrooms can become soggy. To avoid sogginess:
1. Wipe clean, don’t wash them (they absorb water).
2. Pre-broil to release some of the water.
3. In the final stage of baking or broiling, broil or bake briefly, just until topping is browned. Baking too long will result in mushy mushrooms that have released their liquid into your filling. If that happens, the only thing you can do is to carefully drain the liquid, and place the mushrooms on paper towels to soak as much of the liquid as you can.
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