Prepare the grill for direct cooking over high heat (450° to 550°F). Using a damp paper towel, wipe the outside of each mushroom cap to clean. Using a small knife, cut out the stems.
Holding one mushroom in your palm and using a small spoon, gently scrape out all the black gills, being careful not to tear the delicate cap. (It’s important to remove the gills, as they will discolor the filling during grilling.) Repeat with the remaining mushrooms.
Brush a large perforated grill pan with oil. Rinse the chard under cold running water. Stack the wet leaves in the prepared pan.
Grill the chard over direct high heat, with the lid closed, until the greens begin to wilt, 5 to 6 minutes, turning once or twice with tongs. Remove from the grill.
Grill the chard over high heat, with the lid closed, until the greens begin to wilt, 5 to 6 minutes, turning once or twice with tongs. Remove from the grill.
Cut off the stem at the base of each leaf and discard. Fold the leaves in half lengthwise and twist to wring out the excess moisture. Using a heavy, large knife, coarsely chop the greens. You should have a generous 1 cup.
In a medium bowl mix together the chopped greens, feta, mozzarella, ¼ cup Parmigiano Reggiano®, the mayonnaise, scallions, garlic, and nutmeg.
In a small bowl mix together the panko, the remaining ⅓ cup Parmigiano Reggiano®, and the remaining 1 tablespoon oil. Stir 1 tablespoon of the panko mixture into the greens filling. Season the filling to taste with pepper (about ¼ teaspoon).
Divide the greens filling evenly among the mushrooms, then pat gently with your hand to fill to the edges.
Sprinkle the remaining panko mixture evenly over the filling.
Place the mushrooms on the same grill pan. Grill over direct high heat, with the lid closed, until the cheese melts and the topping is deep brown, 8 to 12 minutes. Using a metal spatula and tongs, transfer the mushrooms to plates and serve.
Place the mushrooms on the same grill pan. Grill over high heat, with the lid closed, until the cheese melts and the topping is deep brown, 8 to 12 minutes. Using a metal spatula and tongs, transfer the mushrooms to plates and serve.
