Combine the breadcrumbs, Pecorino-Romano, mozzarella, parsley, mint, garlic, 2 tablespoons of the olive oil, 1/2 teaspoon salt and 1/4 teaspoon pepper in a medium bowl and toss to blend.
Toss the mushrooms with the remaining 2 tablespoons olive oil in a large bowl and arrange on a small baking sheet or plate with the cavities facing up. Divide the breadcrumb mixture among the mushrooms, filling the cavities and pressing down gently to secure.
Place half the mushrooms in a single layer in the basket of a 3.5-quart air fryer. Set the air fryer to 360 degrees F and cook until the filling is bubbling and browned, about 10 minutes. Repeat with the remaining mushrooms.
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